ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

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Transcript of ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

Page 1: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

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_T; | ]!P9RTfOiVB> ;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _&_T; P kt>V eS]9@gT`V@Tu@gT`RTUPVT ;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 7d_T; U . OL= TR V OQ]P@T TYV ]!PTV@OL= TR VOi]!PS`R]!P I-\2IK>UT` ;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N;

_T; UQ;<: k>Veg]T@Tu@T TYV ]P ;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N;

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_T; W-;<: | ]PVB=^OQPV T`>\!^hQOiVB>` ;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 7^_T; W-; | ]PVB=^OQPV T`OQP->6\^hQOV?>` ;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 7

_T; ^ oSf T=RUOQRUT ;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 7c 35J#! %89p 'J+T('87Q89p *

K;<: »^ D >UV eg]T@Tu@gT\!^!R]2]SO ;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; c7PK; »^ D >UV eg]T@Tu@gTwv ^&IS`` Z Y TOL@Th ;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; c2UK; P | ]!P9RT6=?\TPSRTM@T`VD >UV eg]T@T`OiVB>6=^ VO<R!T` ;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; c2U

K; PK;<: ZE>`BIghiV^ VV\7>UP->=^h ;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; c2UK; PK; | ]P9RT=B\!TUPSRUT@T`D >UV eg]T@T`TVV=TUh^fg^V OQ]P ;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; cJWK; PK; P | eg]O<f @TI R] T8RUOiTPVE@gTN= ThQ^&f^V OQ]P ;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; c9_

Page 6: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

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Page 7: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

:7;<: H7I_^ V OQ]P_`hiOQP->^OL= T` ;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; :

:7; | ]7Dw]gOQPS^OQ` ]PS`hQOiPK>^OL=T` ;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; :

:7; P XIVB=TN=TUj-=?>` TUPV^V OQ]Po@GF ISP`BA`yV?C6D T hiOQP->^OL= T ;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; :XW:7; U N^2= OL>V?> hiOQP->^O<=T ;O;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; -::7; W Y A `VBCD TOiPSRU]2D jS^V OL]ghiT ;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N;

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f(t) = x1 + x2t + x3t2 D T`BI-=B>T j_]7I-= t = t1 · · · t5

;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; UJUQ; U =?>6\7= T` ` OQ]PhQOiPK>^OL=T ;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; UJcUQ; W vM=^&D Z Y Re-D O@ V ;O;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; W-:UQ; ^ ` =^PS`[b ]2=?D8^ V OQ]Po@Tn[]7IS` TUeg]!hL@T= ;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; W

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R2 ;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _T:^K; W]7IghiTOIgPSOV?>TUhQhiOQjV OHJIgT ;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _&P^K; P ]PSRV OQ]PTUhQhQOijVOLHJIgTu@gT

R2 ;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _U_T;<: | ]7I-=?]_T`E@TPgOLRT^2I-TUVE\2=^2@OQTUPV ;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _&^_T; jgO<\7=^jgegT TV` TRV OQ]P ;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _&^_T; P ]PSRV OQ]P R]!P9RTfT ;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _&c_T; U ]PSRV OQ]P R]!PSRU^XR!T ;N;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _&c_T; Wa ]PSRV OQ]P R]!P9RTfTO@O*>= TP!VOQ^2]ghQT ;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; K:_T; ^ . OL= TR V OQ]P_`E@T TYV ]PTUV@TI \7=^7@OQTUPV ;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 7P_T;e_ dfhQ^P-V^P-\!TUPV

MTUVV\2=^2@OQTUPV ;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 9_

_T; Y ]hLIV OQ]PS`=?>^hiO`^&]ghQT`TV` ]hLIVOi]!P-]!jV OLD8^hQT ;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 7cK;<: . >RU]2D j4]!` OVOi]!P@gTh^ D8^ VB=ORT ;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; cJW

_

Page 8: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]
Page 9: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

E

: kt>V eS]9@gTu@Twv ^&IS`` ;O;N;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 2 kt>V eS]9@gTu@Twv ^&IS``^XRTRjSO<R!]VjS^&= V OQTUh ;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; 2cP ^RV ]2=O` ^V OQ]P

LU;N;O;N;O;N;N;M;N;O;N;O;N;N;O;N;O;N;M;N;O;N;N;O;N;O;N;O;N;N;M;N;O;N;O;N;N; PK:

U Y I-]S`V OiVBIVOi]!PS`V@OL= TR VT`ªOQPJR!T6=`T` ;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; P

W kt>V eS]9@gTu@T | eg]hQT` A ;O;N;N;O;N;M;N;O;N;N;O;N;O;N;O;N;M;N;N;O;N;O;N;O;N;N;O;N;M;N;O;N;N; _T:

c

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:d

Page 11: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

! " #$% #

&')(*#

+-,.+ /103254769809:;=<);=>@?A:B8DCB098D8FEG8;IHKJL098MCONP69QSRB<T;=>.?A:58MU.>@:569<->V4I098| ]!PS`O@T>= ]!PS`hiT`BA`yV?C6D TN@T >HJIS^ VOi]!PS`9 OiPSRU]PgP9IgT`

2x1 − x2 = 1x :7;<: y

x1 + x2 = 5x :7; y

• WYX[Z.\S]_^@X[ZA]a`cbdZ@XfegX[h.bjikbl]_mk`njopZrq@bfstX[ZuecivXTow]yxz`Z|z~ | e_^2HJIgTM>HJIS^ VOi]!P =TUj-=?>` TUPVTIgPgTN@T=]OiV TO@T hF T`jS^!RT 9 @OLDTPS` Oi]!PS`

R2 ; ^8` ]hLIV OQ]P`T V?= ]7I-RT:9hF OQPV T6=` TR VOi]!P@T`V@gT6ITf@T= ]!OVT`Nx 9 D]!OiP_`EH7I F ThihQT`PST`]!OiTPVYj_^&=^hQh<ChiT`?y ; b­P-]7]V OQTUPV[OROGhQTj_]!OiPVx1 = 2 x2 = 3 R&F T`V 99@gO<=T­hQTORTR V TI-=

x =

(

23

)

@TR2 xR!]OL= #K\2IK= T :2;L: y ;

• SZ@sz]_^.\ZA]a`cbdZ@XfegX[h@bdivbf]_mv`njopZrq@bfstX[ZYecivXSqmkopmv``Z|z~ b­P>R6= OiVhiT`BA`yV?C6D T`]7IS`hQ^ b ]7=BD TORTR V]2= OQTUhQhQT

x1

(

21

)

+ x2

(

−11

)

=

(

15

)

.

hGbÅ^&IgVE@g]PSR­VB=]2IKRT6=YhQT`RU]TROQTUPV`@GF IgPgTRU]2Dw]SOiPS^OQ` ]PDhiOQP->^O<=TN@TN@T6I-f RTR V TI-=`HJIgO!@g]PgPgTPVIgPKVB=]O`OLC6D T~xR!]OL= #K\2IK= T :2; y ;

89j +0)8 1 r PTfT6D jghQTN@T`[A`VBCDT TP@gO<D TUP_`OQ]P P-;2x +y +z = 54x −6y = −2−2x +7y +2z = 9

| e_^2HJIgT>HJIS^V OQ]P = Tj-=?>` TUPV TIgP jgh^P @TmhF T` jS^RUTR3 ]_~aZ~ h1F T` jS^RUT 0xyz

; . TITf jghQ^PS`8` TmR]2ISj_TP!V\7>UP->=^hiTD TUPV`BIgOLRª^PVMIgPST @T=]OiV T ; b­P @OL=^oHJIgThiTjgh^P ^ @O<D TPS`OQ]P_`­TUVh^o@T= ]!OVT/ISPgT @gO<D TUP_`OQ]P ;ofPo\7>UP->=^hGRTV V Tu@T=]OiV TRU]2Igj4T6=^hiTVB=]O` O<CDT jgh^PTP IgPj4]OQPV ` ]hLIV OQ]PoIgPgOHJIgTu@TI `BA `VBCD T ;:7:

Page 12: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

: 6

5

5

−1

X 1

X 2

(2 3)

:2;L: HJIS^V OQ]PS`YhQOQP->^O<=T`

(2 1)

(−1 1)

(1 5)

:2; | ]7Dw]gOQPS^OQ` ]PS`hQOiPK>^OL=T`

Page 13: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

RN :(P

oP @T6ITfOLC6D TOQPV T6=j-=?>V^ V OQ]P ]P RegT=RegThQT`­R] T RUOiTP!V` (x, y, z)@GF IgPgTcRU]2D/]gOiP_^O`]!P-hQOiPK>^OL=Tu@T`

R!TRV T6IK=`YR]!hi]!PgPgT`V TUhQhQTuH7IST6%

x

24−2

+ y

1−6

7

+ z

102

=

5−2

9

.

4%%'(')* 1? )!p9jk8p(')Q"%0)'18J !#"%$'&("*),+.-0/1!23+%45687 9:7 6;<=86'>?3!".>@>A87 0BCDFE!F3"%GH8>@IF".>@!

+-,KJ L 0NM ;I09R 478 0Y; JL<T;=4I>OMY098DCB<A:B8R

n

Rn n ≥ 1 T`V[h1F T` jS^!RTO@T`ER!TRV T6IK=`=?>UTUh`9 nRU]2D j4]!`^PV T` ; b­PKPg]VT6=^

x =

x1;;;xn

hQTNRTR VT6I-=E@TRU]2D j_]` ^PV T`xi i = 1, . . . , n P ` ]OiV x ∈ R

n ; 4%%'(')* gQSR/+-p(8t$7+-#*%')G"%8 T;F;='9U(VXWF)4Y@WF-6Z-0+ [01\4]

Rn 7 !'&B^U 8`_@@a"b?'

ei c i = 1, . . . , n c 8>@C"=daO>@&e;>KC"bf>@C"f"b>F"bOfXN'gUi h $'&(e8_FF"

T`N>h<>DTPV`w@gT8hQ^]S^!`TR^Pg]!PgOH7IST`]!P!V^2IS` ` Of^jgj_Th<>`NRTR VT6I-=`NIgPgOiV^O<=T` ; ` ]2IVuR!TRV T6IK=w@TRnj4T6IgV @]P_R `6F >R6= OL= T@TD8^PgO<C= TIgPSOLHJIgT RU]2D D TIgPgT RU]2Dw]SOiPS^OQ` ]PHhQOiP->^OL= T@T`~RTR VT6I-=`~@T h^ ]S^` TR^Pg]!PgOH7IST ]_~aZ~

x = x1e1 + · · ·+ xnen.

i 89jk+TG"#8 ) jX`k".9l*'m;<n;oeXdX>8"b".>@]b;G@apU?(q.aO>@& &(e>@jg.?I"' 6r:!p.'sd_?,a,"b@K8ONOa?U?9O

t6u3vfuSt wyxmzoF|F~ nDQ8 <n<X<* F ^7@-@OiV OQ]PKTVhQ^/DwIShVOijghQORU^ VOi]!P-j_^&=IgP` R^h^OL= T`]!P!VYhQT`E@TITfD]j*>6=^ VOi]!PS`@TO]S^` TM@S^PS`IgPKT` jS^RUT

R!TRV ]2=OQTUh ;TY ]OiVxTV

y@TITf R!TRV T6IK=`@T

Rn ]!PK^

x + y =

x1 + y1;;;xn + yn

.

Y Oα ∈ R ]P ^

αx =

αx1;;;αxn

Page 14: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

:6U 6

tu3vNu3v F D~%Q8C| |~ !TDj-= ]T@TISOV ` R^h^OL= T8T`yV IgPgTD^jgjghQOQR^ V OQ]P ]SOihQOiPK>^OL=T @T

Rn × Rn @g^PS`RPg]V?>UT

< ·, · >]7IIjghLIS`R]7D DwIgP->D TUPV

xT y%

xT y = x1y1 + · · ·+ xnyn.TjK= ]T@TIgOiV[` R^h^OL= T

xT yT`yV@]!PSRMIgP=B>TUh ; Tj-=]T@TIgOiV[` R^h^OL= TT`V`[A9D >V?= OHJIgT

xT y = yT x.

d]2I-=x y TUV z

@gT`ER!TR VT6I-=`@TRn TV α ∈ R

]!PK^ %xT 0 = 0, xT ei = xi, xT (y + z) = xT y + xT z, (αx)T y = α(xT y)

i 89jk+-XG"%8 g !".9k'DU`&'&(DX>8"b"3U>? ;=>8NH_@@a"b?NC0 ∈ Rn ?"o;> #a'kOX

0

54%''1Q L~X%*Q%+-01'F I"A88j8? _@@a"%@'x

@"y

8Rn >@C"d>?"3rX>,lF>@*FS

xT y = 0

tu3vNu |8*~SC<r PSTuD8^ VB=ORT

m× nT`VIgPV^&]ghQT^2It@Tc`RU^hQ^O<=T`=^P-\7>`YTP

mhQO<\!PgT`TV

nR]!hi]!PgPgT` ; T`D ^VB=OQRUT`` T6=?RTUPVj-=OiP_ROQjS^hQT6D TUPV9 = Tj-=?>` TUPV T=@T`VB=^PS`[b ]2=?D8^ VOi]!PS`hiOQP->^O<=T`@T

Rn @g^P_`

Rm ; b­PKPg]V T6=^ aij hQT`4>UhL>6D TUPV`@T­hQ^wD ^VB=OQRUT

A i = 1, . . . , m j = 1, . . . , n; »^uD8^ V?= ORT

A= Tj-=?>` TUPV T[h^V?=^PS`[b ]2=?D ^V OQ]PhQOiP->^OL= TN@T

Rn @g^PS` Rm `[ISO<R ^PVT&%∀x ∈ R

n, y = Ax ⇔ y ∈ Rm TV

yi = ai1x1 + · · ·+ ainxn, i = 1, . . . , m.

Y O]PFPg]VTA•j j = 1, . . . , n hQT`cR]hQ]PSPgT`N@T A

TUVAi• i = 1, . . . , m

hQT`NR!TRV T6IK=` hQO<\!PgT`w@TA h^VB=^P_` b ]7=BD8^ VOi]!PDj4T6IV^&IS``OG`F >R=O<=TN@T`E@gT6ITfoD ^PgOLC6=T`Y`[ISO<R ^PVT` %

y =

n∑

j=1

xjA•j]2I

yi = ATi•x, i = 1, . . . , m.

2 IS^PK@m = n ]PKjS^2= hQTN@TND8^ VB=ORTRU^2=B=?>UT ;

8Jj +%018 g Y ]!OV[h^ D ^VB=OQRUTA@]PgPK>UT jS^&=

A =

[

2 −11 1

]

.

TNR!TRV T6IK=x@]PVh1F OLD ^2\T jS^2=

AT`V [ 1

5

] T`yV` ]hLIVOi]!Po@TI `BA`yV?C6D T

2x1 − x2 = 1

x1 + x2 = 5

| F T`V@g]PSR hiT RTR V TI-= [ 23

] V?= ]7I-R2> jghLIS`KeS^&IV ; b­P=TV?= ]7I-RT `BI-=h^ #K\7I-= T :2; P hQT`KOiP-b ]2=?D ^V OQ]PS`R]7D DwIgPgT`^&ITf #K\2I-=T` :7;<: TUV :2; T;

Page 15: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

RN :XW

(2 3)

(1 5)

A

:7; P DXVIgVB=TN= Tj-=B>`TPV^ VOi]!P @TI`BA `VBCD T x :2;L: y Zzx :2; y

Y ]!OVA B @T`OD8^VB=OQRUT`

m × n ` ]OiV x y @gT`ORTR VT6I-=`O@T Rn ]!P´^KhQT` j-=]jK= OL>V?>`O>UhL>6D TP!V^OL= T``BIgOLR ^PV T` %A(x + y) = Ax + Ay

(A + B)x = Ax + Bx

αAx = A(αx), ∀α ∈ R.

»^uD8^ V?= ORT­OL@TPV OiVB>InT`yVhQ^wD ^VB=OQRUT­R^&=?=B>T@]PVhiT`RU]hQ]PgPgT` b ]7=BD TUPVhQ^w]S^` T­R^Pg]!PgOLHJIgTM@T

Rn ;b­P^ @]P_RInx = x, ∀x ∈ R

n.r PgTND8^ VB=ORTN@O^&\]!PS^hQT Pg]VB>T A = diagd1, . . . , dn RU]2=?= T`j4]PK@9 IgP ReS^P-\!T6D TUPVE@GF >RegThihQT ;

4%%'(')* [35+T6'1$28t+-#p+ *p98F ]#;!;D& ".K3aH".?!b;=>?G#m7 XD& ".K3a G'9UA

U& F".K.aO

AT "bHyT (Ax) = (AT y)T x, ∀x ∈ R

n ∀y ∈ Rm.

x :7; P y hST`VS]gOiTP8TUPV TPK@TI~HJIgT[hQTj-=T6D OiT=qj-=]9@-IgOV` R^h^OL= T@g^P_`Vx :7; P y T`V @g^PS`

Rm TVhQT` TR]!PK@ @g^PS`

Rn ; b­PD ]!P!V?= Tl^hQ]2=`VHJIgTch^ D8^ V?= ORT

AT ^8j_]7I-=hiOL\PgT`[hQT`R]!hi]!PgPgT`@TAx TUVM@]!PSRj4]2I-=R]!hi]!PgPgT`hiT`hQOL\PgT`

@gTAy ; b­P =T6D8^&=zHJIgTOH7ISThQ^Pg]V^ VOi]!Po@S^PS`Nx :2; P yT`VR]!e->6=TUPV T ^XRTR RTUhQhQTO@-IKjK= ]T@TIgOiV`RU^h^OL=T ;

4%%'(')* qQ[35+T6'1$28np)!jkJ6'1G"#8F 0X& F".K.aOa?KG"6I"b &(G?".K.SAT = A

q>@jU>?K

aij = aji

t6u3vfu FD ~% |!~S\~S<Tcj-=]T@TIgOiVD8^ V?= OROQTUh R]2=?=T` j_]!PK@ 98h^DR]7D j_]`OiV OQ]Pt@T @T6I-f ^jgjghQOQR^ VOi]!PS`hQOiP->^OL= T` ; b­P VB=]2IKRTh 9hQTj-=T6D OQT6=RU^hQR6Igh`B>6=OiTITf @TIj_]!OiPV@TOR9IgTu@Th^ R]7DjShiT6f OiVB>0%

C = AB −→ cij =

n∑

k=1

aikbkj ∀i, j x :7; U y

Page 16: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

:^ 6

54%''1Q 3+? #;F;='9UnZ 7 >#;G'F".>@TGUG& o"IOH7 B;CO>F!I" #a'kO c Cd&(". ;C9.a"3U>?jI_@Ue7 XdFI".>@ @"7 X!^".I"SF"3U>? r P S]j=T6R OiTPV@]PSR:9 DTUVV?= T 9 y]7I-=VIgPSTc`]7D DT

sTUPhLIgO =^ y]7IV^PVYhQTj-=]T@TIgOiV@Tw@gT6ITfKPg]2D/]-= T`

aTVb F]_~aZ ~

s := s + a ∗ b.TRU]VV@GF IgPj-=]T@TIgOiV` R^h^OL= TO@TN@T6I-f RTR VT6I-=`E@T hi]!P-\2IST6I-=nT`V@]PSRO@T

nS]j_` ;X#RTRcRTV V T @T>/#SPSOVOi]!P hQTPg]7Dw]-=Tw@GF ]j*>6=^ VOi]!PS`P->RT` `^OL= T`j4]2I-=­T ¦TR V?IgT6=hQTj-=]9@-IgOV x :7; U yYT`VM@ThF ]7=?@T=Tw@T

n3 S]!jS` ; b­PRT=B=^ jghLIS`[hQ]OQPnHJIgTcRUTlR]V VB=?C`>UhQT6R2> T`yVTPbÅ^OiV>(HJIgO<R ^hiTP!V^&ImR]VM@gThQ^=?>` ]hLIV OQ]P@GF IgP `[A`VBC6D T hQOiP->^OL= T:9nOQPSRU]PgP9IgT` ;d]2I-=hQTj-=]T@TIgOiVD8^VB=OQRUOiThG]PhQ^j-=]jK= OL>V?>`BIgOLRª^PV T

(AB)T = BT AT .

54%''1Q 6 7Q8Jp8F D7 C_?'''m7 XD& "3p3aea'pOG# c X>8"%G#A−1 c "='7 F3D& ".K3a q>@'

07 '9Um 8k"bKse"%'9UeAA−1 = A−1A = I.

SF >V?IK@T @Th1F TfO`yVTUPSRUT @Th^ D8^ V?= ORTlOiP9RT=` TcT`yVOIgP j-= ]7]ghLC6D TlRUh<> @Th1F ^hL\2C]-= T/OQP->^O<=T/HJIgOq` T6=^VB=^OiVB> ^2I ReS^jgOV?= T T; b­P RT=B=^mTP>jS^2=VOQR6IghiOQT6= j4]2I-=zHJIg]OYhQT`V?=^PS` b ]7=BD8^V OQ]PS`l`OQP-\2IShiOLC6=T`P!F ]PV jS^`@GF OiP9R!T6=`T ;+A, G?AR58c098F2 <eMu0 09:A09:5C5476 2 <)4 RB:'09:B80KJBU@0 CB0A0NM ;=09R54I8r PI` ]2I_` ZwT` jS^RUT RTRV ]7= OQTUhS@T

Rn T`yV/IgP TUPS` T6D/]ghQT~b T6=?D >8j_]7I-=chQT`]!jQ>=^V OQ]PS`u>UhL>6D TUPV^O<=T`N@!F IgPT` jS^!RT hiOQP->^OL= T h1F ^2@-@gOVOi]!P-TUV[hQ^ DwIShVOijghQORU^ VOi]!PjS^&=EIgP `RU^hQ^O<=T ;SF TUP_`TDw]ghQTM@gTV]2IVT`hiT`YR]2D/]gOQPS^O`]!PS`hiOQP->^O<=T`@GF IgPDTPS` T6Dw]ShiTS@TOR!TR VT6I-=`@T

Rn T`VYhQT `]7IS` ZT` jS^!RT TUP-\!TUPK@-=B> jS^2=S Pg]V?> linS %

linS =

x ∈ Rn | x = α1x

1 + · · ·+ αnxn, αi ∈ R, xi ∈ S, ∀i = 1, . . . , n

.b­PR2>= O$#STbÅ^ROQhQT6D TUPVHJIgTlinS T`yVV]SOiTP IgP ` ]2IS`[ZwT` jS^RUT­TUVH7IST 0 ∈ linS ;dS=TUPg]!PS`VISPnRTR VT6I-=

u1@T

R2 j_^&=T6fT6D jghQT u1 = (2 1)T ; F TPS` T6Dw]ShiT @T`RTRV TI-=` y

@TR

2 V ThQ`HJIgTy = αu1 ] α

T`yVIgP=B>TUh%H7ISTUhR]PQH7IST T`yVIgPm`]7IS`[ZtT`jS^!RTu@T R2 hiTc` ]2I_` ZwT` jS^RUT linu1;Q` ]2IS`` T`V>h<>D TUPV`­j_TI-RTPV,V?= T/@T>R6= OiV`jS^&=­hiTc` T6ISh»jS^2=^2D CVB=T

α ]!Pn@OiVMHJIgTh^ ]a\Z.`V]_mv` @T linu1T`yV

>6\^hQT:9 :7;u1T`yVhQTN\2>P->6=^ VT6I-=E@T

linu1;Z[^! y]2IV]PS`D8^OQP!VTUPS^PVMIgPF^&IgVB=T/R!TR VT6I-=

u2 = (−1 1)T ; T8` ]2IS`[ZwT` jS^RUTTUP-\!TUPK@-=B>jS^&= u1, u2T`V[hF T`jS^!RTR2 V ]7IVTUPV OQT6= HJIgOGT`yV@Tu@gO<D TUP_`OQ]P T;

i 89jk+-XG"%8 L (X>@& ^?Oj8(lFG'XG'F"b?Kd;=@F" @"3A;CIAl'\Q(Uy:&('*S>@! 6_@@au3 =

(4 2)T c UD>8"bb;o8aODl\OGn;o u1, u3O#"7 !'&B^Ue8O

y"b9d

y = αu1 + βu3 = (α + 2β)u1.j7 >$#

linu1, u3 = linu1=" H&(!S>@

b­P ^ R9I HJIgThiTPg]7Dw]-=T@T\2>UPK>6=^ V TI-=`cT`yV`BIgjQ>= OQT6IK= 9 h^ @O<D TPS`OQ]P ` ORT=V^OQPS` \2>UPK>6=^ V TI-=`^jSjS^&= V OQTUPgPSTUPV^&I`]7IS`[ZtT`jS^!RTTUP-\!TUPK@-=B> jS^2=[hiT`[^&IgVB=T` ; b­P@gOVHJI!F Oih`[` ]PV ow]a`hrik]aX[Z@\GZ.`cb hdeZ.` iv`b% ;. ^PS` hQTR^`R]!P!V?=^O<=T hF TPS`TDw]ghQT @T`M\2>P->6=^ VT6I-=`[T`yV ow]a`ghriv]aX[Z.\Z.`cb]a` hdecZ@` iv`b TV` TlPg]7D DT'& i(.Z@TI`]7IS`[ZtT`jS^!RT ;Jr P-` ]2IS`[ZtT`j_^RT­j_]` `BC@gT­TP \2>P->6=^h*IgPgT OQP #SPgOiVB>N@TO]S^!`T` D8^O`V ]2IgV T`RUT`]S^!`T`f]!P!Vh^ D6,D T R^&=z@OQPS^hQOV?> h^' ]a\GZ@`)r]_mk` @TI ` ]2IS`[ZtT`j_^RT ;T~=?>`BIghiV^V`BIgOLR ^PVOb ]2I-=PgOiVuIgP D]A!TUP´j-=^ V OHJIgT~@T V T`yVT6=c` OkRTR V TI-=`` ]PVhQOiP->^OL= TD TUPVOiPK@->Zj4TUPK@g^PV` ;

Page 17: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

7 :_ + *p')')*1

k_@@a"%@'D8

Rn c v1, . . . , vk>@C"=9XG#?9& C"9\8GS;\X"INS c ?"6'?U&('C"S c

α1v1 + · · ·+ αkvk = 0 ⇐⇒ α1 = · · · = αk = 0.

hT`VRh^OL=HJIgTV]2IV TUP_`TDw]ghQT/@TjghLIS`M@TnR!TRV T6IK=`M@T

Rn T`V­hQOiP->^OL= TD TUPVM@T>j_TPK@g^P!V RT/HJIgOD ]!P!V?= T/H7ISTch^ @OLDTPS` Oi]!Pn@GF IgP ` ]2IS`[ZtT`j_^RTu@gT

Rn T`VIgP TUPV OQT6=MR ^&=OQ^P!V@gT d x `]7IS`[ZtT`jS^!RTP9Igh 0 y9nxRn h<ISO Z D0,6D T(y ;r PgTD8^VB=OQRUT

A9

mhiOL\PST`TV

nRU]hQ]PgPgT`S=TUj-=?>` TUPVTVIgPgTVB=^PS`[b ]2=?D8^ V OQ]P hQOiPK>^OL=T@T

Rn @g^PS` Rm ;d]7I-=YV ]7IVx@T

Rn ]P@T>/#_PgOV[hF O<D8^&\!TN@T xjS^&=Yh^cV?=^PS`[b ]2=?D ^V OQ]P

ARU]2D D T hiTNR!TR VT6I-=

y@T

Rm V TUhHJIgT

y = Ax =n∑

j=1

A·jxj , A·j ∈ Rm, j = 1, . . . n.

T-` ]2IS`[ZwT` jS^RUT @TRm TP-\TPK@T=?>jS^&=hQT`R]hQ]PSPgT`/@T

AT`V ^jgj4TUhL> .mvs PZ lecizqrZB]a\Gi.x Z @T A PS]VB>

Im(A);%| F T`Vu@]P_RchF TPS` T6Dw]ShiT~@T`

y@T

Rm HJIgO`F >R=O<R!TUPV` ]2IS`h^ b ]2=?DTAx

j4]2IK=MISPFRUT6= V^OiPx@T

Rn ;i 89jk+TG"#8 $ i +- L"#%8jk+TX'1$28F !>@I"

AXN& F".K.aO

m×n N'l 8

A c X>8"%Grang(A) cO#"*UX>@& ^?O& & F8fa>@>@!X68

A9XGOO'&(X"!\G ;='\C"b m7 O#"CKS:UX>@& ^?O& &

Dl:XOf9XG#?9& C"09\8GS;\?C"% jB>@\a

rang(A) ≤ minm, n.. ThQ^~@->/#SPgOiV OQ]Pj-=?>R6>@TPV T ]!PDVO<=T hiTN=?>`BIghiV^Vb ]!PK@g^2DTPV^h`BIgOLR ^PV

dim (Im(A)) = rang(A).

+-, EG8;=HKJL098DC N69QRB<T;=>@?-:58MU@>.:569<A>@4I098 5?-JL? AH9:5098 4%%'(')* ? )!p9jk8 %Qj X9#8F 0(%!"b$& 7 G@:"3U>?!69CGO?9O O#"n3"6ZZ8/-n4Dp!,aO>@\ &('& ^?dO#"C

Y ]!OVAh^D8^ V?= ORT

m × n@-IF`BA`yV?C6D Ttx)@g]PSRjS^!`P->RUT`` ^O<=T6D TP!VRU^2=B=?>UTXy ; X[P_^hLA ` ]P_` h1F TUPS` T6D/]ghQT@gT`` ]hLIV OQ]PS`E@-I`BA`yV?C6D TN@GF >HJIS^ VOi]!PS`YhQOiP->^OL= T`eg]2D ]2\7CUPgT`

Ax = 0.x :7; W y

hT`VRUhQ^O<=VHJIgT d T`yVYV ]7I y]2IK=`` ]hLIV OQ]P ; + *p')')* g 7 !'&B^? 8'>?3!".>@!H!B !#"%$'&(d99XGOO q #smO#" '>bb;oaA_?,a"%>@KU

Rn 6'>bb;oa O#"n;F;='G-6ZoWo1 8A c X>8"%G

ker(A)

oP T4TUV ` O x TUV y`]!PV` ]hLIVOi]!Po@T~x :2;eW y x + y

T`yV^2IS``OG` ]hLIV OQ]P ^OiP_`O HJIgT αx α ∈ R;

. F ISPgTVD8^PgOLC6=T\2>P->6=^hQT hF TPS`TDw]ghQTE@T`SRTRV TI-=` ]7=Veg]2\!]PS^2ITf 9ISP ` ]2I_`d TUPS` T6D/]ghiT S@TRTR V TI-=`T`yVVISPK` ]2IS`[ZwT` jS^RUTR!TR V]2=OiTh Pg]VB> S⊥ ]_~aZ ~

S⊥ =

x ∈ Rn | xT s = 0, ∀s ∈ S

.

b­PoR2>= O$#ST=^ HJIgTS⊥ = (linS)⊥ ;b­PF`6F ^j4T6=]!OVN@]!PSR H7ISTlV]2IVc` ]2I_` ZwT` jS^RUTwR!TR V]2=OiThj_TIV ,V?= T = Tj-=?>` TUPVB> @T~@T6I-f D8^PgOLC6=T`O@O`[ZVOiP_R V T`:%

Page 18: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

: 6

R]2D D T[TPS`TDw]ghQT@gT­R]7Dw]gOQPS^OQ` ]PS`hiOQP->^O<=T`4@GF IgPPg]7Dw]-=T#SPgO@TRTRV TI-=`x T`jS^!RT@gT`4RTRV TI-=`R]hQ]PSPgT`]2IOLD8^&\TO@Th^ D ^VB=OQRUTMb ]2=?D >T jS^&=[RUT`R]!hi]!PgPgT`?y ;R]2D D TTPS` T6Dw]ShiTM@T``]!h<IgV OQ]PS`@GF IgPD`BA `VBCD ThQOiP->^OL= T­eg]2D ]7\2CUPSTwx Pg]A^&I @T­hQ^wD8^ VB=ORT@g]PVhQT`hiOL\PgT`R]!P!VOiTPgPgTP!VhQT`[RU] T ROQTUPV`E@gT`>HJIS^V OQ]PS`zy ; 8Jj +%018 L | ]P_`O@T>6=]P_`N@g^PS`

R3 hiT8jgh^P L@T`NRTR VT6I-=`u@]!P!Vh^-VB=]O`OLC6D TRU]2D j_]` ^PV T

x3T`yVP9Igh ;

LT`V@]PSRhF TPS` T6Dw]ShiTN@T`[`]!h<IgV OQ]PS`E@T hF >(HJIS^ VOi]!P-eg]7D ]2\2CPgT

x3 = 0 ]2I]gOQTUPL = x ∈ R

3 | Ax = 0, A =[

0 0 1]

.

| T` ]2IS`[ZtT`j_^RTOR!TR V]2=OiTh¦T`V­Rh^OL= TD TUPVV@Tu@OLDTPS` Oi]!P TUV[j_TIV ,V?= TTP-\TPK@T=?>jS^&=V@TITfo@gTc`T`>UhL>ZD TUPV`hQOiPK>^OL=T6D TUPVOQPK@T>j_TPK@g^P!V`Ox Pg]!P-R]!hiOQP->^OL= T` @g^PS`RUTR^`zy ; b­PDj-=TUPQ@jS^2=T6f TD jghiT­hiT`RTRV TI-=`(1 0 0)T TV

(0 1 0)T ;. ]!PSR

L = x ∈ R3 | x = Bz, z ∈ R

2, B =

1 00 10 0

b­P~=T6D8^&=zHJIgTEH7ISThiTVR!TRV T6IK=qhiOL\PgTV@gT[hQ^OD8^VB=OQRUTAT`V]2= V eg]7\]!PS^h 9V ]7IV R!TR VT6I-=4@T

L; b=RUTVRTR VT6I-=@T

R3 TUP-\!TUPK@-= TMIgP`]7IS` ZwT` jS^!RTR!TRV ]2=OQTUhG@TO@O<D TPS`OQ]P : hF OLD8^&\!TO@gT AT ; b­P@OL=^^hQ]2=`4HJIgThF OLD8^&\T@TAT xÅ` ]2IS`[ZtT`j_^RTMRTR V]2=OiTh@gT`hQO<\!PgT`zyT`V[]2= V eg]7\]!PS^hQT­^2I-Pg]A^&I ;

+4Qp('(')*L !>?3"A

XD& F".K.aOm× n

]

Ker(A) ⊥ Im(AT ),!

Rn

Ker(AT ) ⊥ Im(A),?!

Rm.

| T`j-= ]!j-=O<>UVB>`\7>U]7D >UVB=OLHJIgT`GR]!PSRT= PS^PVhiT`GjS^OL=T`G@Tf` ]2I_`d T` jS^RUT`4]2= V eg]7\]!PS^&ITf` T6=]PVGTfjghQ]OiVB>T`^&I ReS^jgOiVB=T UQ;T`E@OLDTPS` Oi]!PS`= T`j4TRV OLRT`E@TRT`H7I_^ VB=T `]7IS`d T` jS^RUT`PgTu@T>j_TPK@TUPVHJIgTu@TI=^P-\ @T

A;

dim Ker(A) = n− rang(A), dim Im(AT ) = rang(A),@S^PS`

Rn

dim Ker(AT ) = m− rang(A), dim Im(A) = rang(A),@g^PS`

Rm

8Jj +%018 $ Y ]!OV[h^ D ^VB=OQRUT2× 2

`[ISO<R ^PVT

A =

[

1 −3−2 6

]

.

b­P ^rang(A) = 1

;,SF T` jS^RUT RU]hQ]PgPST`]2I

Im(A)RU]PV OQTUPVYV ]7IS`YhQT`ED/IghVOijShiT`V@TIRTR V TI-=

(1 − 2)T ;,T Pg]A!^2I @TARU]PV OQTUPVhQT`ED/IghVOijShiT`E@T

(3 1)T ;,SF T` jS^RUThQOL\PgT`Y]2IIm(AT )

R]!P!VOiTPVYhQT`D/IghiV OQjghiT`V@T(1 − 3)T ;,T Pg]A!^2I @T

AT R]!P!VOiTPV[hiT`DwIghiV OQjghQT`V@T(2 1)T ;| T`EHJIS^VB=T`]7IS`d T` jS^RUT``]!PV@T`E@T=]OiV T`E@gT

R2 ;KY O]!PKRe_^P-\!T­h^ @T6ITfOLC6D TR]!hi]!PgPgTTUP (−3 7)T hQT`R]!hi]!PgPgT`q`]!P!V^hi]7=`hQOiPK>^OL=T6D TUPVqOiPQ@T>Uj4TUPK@S^PV T`TVrang(A) = 2

;!. ^PS`RUT[RU^!`Im(A) = Im(AT ) = R2TV

Ker(A) = Ker(AT ) = 0 ;

Page 19: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

1 :(c+-, O47<A:B8v?)4 JL<T;=>@?-:58 69U@6KJL09:;=<A>V47098D8R54 U.098 JL<T;47>OMY098 4%%'(')* Q;F; "3'!Sg3>?p& F".>@yG'G'&('C"IO(8Oe9 lXjq9OI;=,a,".I_@& C"d8 a>@>@!XSs=7 CD& F".K.aOHXHa>?&B^?9*9>@ 8d"3>?9d"3'!Sg3>?p& F".>@!#3_O?C"%

h &(I". ;C9.a"3U>?T=7 CDl:X q9OI; \a>@>@!XKsn;oj#a'kOX>@ Ch 8F:3"3U>?TH8@Bl:XOAqOb; Xa>?U>?FXO sh ;p&BF"IF".>@ 8e8?B9 lXAq9OI; Xa>@>@!XSs

T!~SD\~ xD|= Fz<8 |0Fx O y r PST[VB=^P_` b ]7=BD8^ VOi]!P/>h<>D TUPV^O<=TPgTReS^P-\Tj_^`hQTV=^P-\u@!F IgPSTVD8^ VB=ORTwx R&F T`V IgPgTV?=^PS`[b ]2=?D ^V OQ]PPg]P ` OiPK\2IghQO<C= TXy ;x OQOy r PgT8V?=^PS`[b ]2=?D ^V OQ]P >UhL>6D TUPV^OL= T`BI-=hQT`chiOL\PgT` x=T` j ; RU]hQ]PgPST`zyM>HJIgOLRª^2IV 9D/IghiV OQjghiOQT6=chQ^D8^&ZVB=OQRUT69 \!^2ISRegT x)= T`j ; 9 @T= ]!OVT(yjS^&=MIgPgT D8^ V?= ORT/>UhL>6D TUPV^O<=T]7]V TPJISTlTP ^jSjghiOHJIS^PVhQ^ D6,D TVB=^PS`[b ]2=?D8^ VOi]!P9hQ^ D8^VB=OQRUTO@TP!VOV?> ; 89j +0)8 q

A =

1 0 0 21 2 0 12 1 3 4

Y O4]!PD^2@K@OVOi]!PgPgT­hiT`4@TITfj-=T6D OLC6=T`hQOL\PgT`fTUVHJIgT­h1F ]P-OiPS`R=OLRT[hQT=?>`BIghiV^ VY`[IK=h^j-=T6D OLC6=ThQO<\!PgT­`^P_`Re_^P-\!T6=YhiT`E@T6I-fK^&IgVB=T` ]!P]2]VOiTP!V

A′ =

2 2 0 31 2 0 12 1 3 4

b­PKj4T6IVR7>6=O #_T6=EHJIgTA′ = EA ^XRTR

E =

1 1 00 1 00 0 1

~%n68|6 .D CnSn | F T`VIgPgTl`[>(H7ISTUPSRUTO@gTVB=^PS`[b ]2=?D8^ VOi]!PS`>UhL>6D TUPV^O<=T`EHJIgOVB=^PS`[b ]2=?D TOIgPgTcRU]hQ]PgPgTN@GF IgPgTwD ^VB=OQRUTTP R!TRV T6IK=EIgPgOiV^OL=T x R ; ^ ; @ ; @Th^ ]S^` T ]2= V eg]!Pg]2=?D >UT@gT

Rm y ; SF >UhL>6D TUPVHJIgO!@]!OV` TVB=^PS`[b ]2=?D T6=TUP :T`yV hQT eg]mkb ; dq^&= T6f TD jghiT ` ]OiV Ash^DR]!hi]!PgPgT09jgOLR]V T=­TUV

arshQTljgOLR]V x

ars 6= 0y ; b­P T4TR V?IgT~@g^PS`hF ]2=z@T= T DwIghiV OQjghQOiT=[h^lhQOL\PgT

rjS^&=

1/ars j_]7I-=V ]2IgVi 6= r = TDjShQ^!RT6=fhQ^chiOL\PgT i

jS^&=hQ^l`]7D DTM@Th^hQOL\PgTiTUV@gTh^hQO<\!PgT

rDwIShVOijghQOL>UTjS^&=

−ais;

89j +0)8 3 dfO<R!]V ]!PS`Yh^ljK= TDOLC6=TR]!hi]!PgPgTO@ThQ^ D8^ V?= ORT`BIgOLRª^PV T

A =

1 1 11 0 −1−2 1 3

Page 20: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

&d 6

^XRTRa11 = 1

RU]2D D TjSO<R!]V ; b­P@]!OV@]!PSRMDwIghiV OQjghQOiT=A9 \!^2ISRegTjS^&=h^ D ^VB=OQRUTO>UhL>6D TUPV^O<=T

E =

1 0 0−1 1 0

2 0 1

TV[hQTN=?>`BIghV^ VT`VA′ = E · A =

1 1 10 −1 −20 3 5

h4T`VRh^OL=4HJIgT ` O_V ]2I_`fhQT`jgOLR]V`f`BISRURUT``O<bÅ`f` ]PVPg]PP9Igh` IgPST­`B>HJIgTUP_RT@T n

jgOLR]V^&\!T`T4TR V?I->``BI-=hQT`nR]!hi]!PgPgT` @Th^ND8^ V?= ORT^XR!TRYhQTjgOLR]V`BI-=fh^w@O^&\!]PS^hiTYV?=^PS`[b ]2=?DT=^h^ND8^ V?= ORT[TPhF O@TUPVOV?> ;

Y OEi

T`VhQ^ D8^ V?= ORTO>h<>DTPV^OL=T^!` ` ]ROL>UT ^&IjgOLR]V^&\!TM@T h^iJC6D T R]!hi]!PgPgT ]!PK^

In = EnEn−1 · · ·E1A

@GF ]IgPgTjK= TDOLC6=TMD >UV eg]T@Tj4]2IK=[RU^hRIShiT=h1F OiP9R!T6=`TO@GF IgPgTND8^ V?= ORTA−1 = EnEn−1 · · ·E1.

. ]PSR­j_]7I-=RU^hQR6IghiT=fhF OQP9RT=` T@GF IgPgTMD8^ V?= ORT OQhG`[I VV@GF T ¦TRVBIgT=hQT`jgOLR]V^&\!T`fTUPDj_^&=^hQh<ChiT`[I-=h^uD8^ZVB=ORTOL@TPV OiVB> ; »^uD8^ V?= ORT19cOQP9RT=` T6=`TVB=^PS`[b ]2=?D Tj-=]2\7= T` ` O<R!T6D TUPVTPDh^wD8^ V?= ORT­OL@gTUPV OiVB>^hQ]2=` HJIgThF O@TUPVOV?>O@gT6R OiTPVYhF OiP9RT=` T ;TY F Oih¦T`yVER9=^O*HJIgT H7I_^PK@K^&ISR6IgPDjSO<R!]VYPJISh4P F T`yVE=TUPSRU]PVB=?> RTUVY^h<\!]2=OVe-D T^/IgPgT R]2D jghQTfOiVB>M@TO(n3) Oih¦j4T6IV^ V V TUOQPK@T=T O(2n3)

@S^PS`hQT­RU^!` \7>UP->=^hSTUVP->RT` ` OVT­hQT­`V ]R ^&\!T@T@TITf D8^ V?= ORT`

n× n; b­P-hLIgOGj-=?>b>= T=^chiT`4D >UV eg]T@T`>VBIQ@O<>T`^2I-ReS^jgOV?= T `BIgO<R ^P!V ]S^`B>UT`ThihQT`^2IS``O`BI-=V@T`[jgO<R!]V^&\T`Y`[ISRRT` ` O bÅ`@Th^ D8^ VB=ORT D8^OQ`YjghLIS`E=]2]KIS`yVT`YTUVVD]!OiP_`R]gV T6I_`T` ;

+A, E8F;=HKJL098MC N69QR5<);=>@?A:B8DU@>@:B69<A>V47098Y ]!OV

AIgPgTD8^ VB=ORT

m×n@]PVfhQT`hiOL\PST`fRU]PV OQTUPSPgTUPVhQT`R] T RUOiTP!V`4@-ID`[A`VBCDThiOQP->^O<=T[TUV

bIgP

RTR VT6I-=@TRm R]!PV TUP_^PVfhQT­` TR]!PK@ D T6D/]-= T ; b­PReST6=RegT9w@->R=OL= ThF TPS` T6Dw]ShiTM@T`

x@T

Rn ` ]hLIV OQ]PS`@TI `BA `VBCD TAx = b.

x :2; ^ yX#Rª^PVM@TPS]2IS`j_]`T=hQTj-=]2]ghLC6D Tw@Tch1F TfO`yVTUPSRUTw@GF IgPgTl` ]hLIVOi]!P D ]PV?= ]!PS`VHJIgThF TPS` T6Dw]ShiTw@gT``]!h<ITZV OQ]P_` HJIS^PK@-OihP F T`yV[jS^!`R OL@T T`VVIgP]2] yTUVV@T Rn @!F ^!`j4TRVbÅ^2D OihQOiT= ;Y ]!OV

x0 ISPgT`]!h<IVOi]!PcjS^2=VOQR6IghQO<C= T@TI `[A`VBC6D TNx :7; ^ y ; Xhi]7=` V ]2IgV T`]!h<IgV OQ]P @TI`BA`yV?C6D Tj_TIVf`6F >R6= OL=T` ]2IS` h^ b ]2=?DTx = x0 + y y ∈ Ker(A)

; ofP´T4TUVcR]7D D TxTV

x0 ` ]PVN@T6I-f´` ]hLIV OQ]PS`O@TI `BA`yV?C6D T ]P´^A(x − x0) = 0

; . ]!PSR hF TPS` T6Dw]ShiT @T`` ]hLIV OQ]PS` `F ]7]V OQTUPV jS^&=MIgPgTlVB=^P_`h^ VOi]!Pn@TI ` ]2I_`d T` jS^RUTPg]A^&I@TA;Q hTUP ^ @]P_R hQ^ b ]7=BD TO\2>]2D >V?= OHJIgT TVHJIS^PQ@

bR ^&=OiT ]P R]!PS`VB=?IgOVVISPgTObÅ^&D OQhihQTu@GF ]2] yTUV`hQOiP->^OL= T`jS^&=^hQh<ChiT`TUPVB=T­TITf x)R]OL=#K\2IK= T :2; U y ; b­P-hQT`^jgj_ThihQTN@T` vivXV]_h.bdh )ow]a`ghik]aX[Z ]7I .mvs PZ lecizqrZ i`Z ;*Y OGISPgTNR ^&=O<>UVB> hiOQP->^OL= T jS^!` ` T­j_^&=hF ]7= OL\OQPgT R2F T`yVEIgP ` ]2IS`[ZwT` jS^RUTR!TR V]2=OiTh ;WOiTPH7ISThQ^~@gO<D TUP_`OQ]Pm^OVV>UVB>w@T> #SPgOQTj4]2I-=@->R=OL= T h^ \2>UPK>6=^ V OQ]P@GF IgP ` ]2IS`T`j_^RTMRTR V]2=OiTh ]!PhF TDjShi]!OiTj_]7I-=OIgPgT R ^&=O<>UVB>chQOQP->^O<=TcjS^2=^PS^hQ]2\!OiTcj_]7I-= Tfj-=O<D T6=hQTlPg]7Dw]-=T/@gT @T\2=?>`@TlhQOL]_T=V?> ; Y O

V = x ∈ Rn | Ax = b T`yVIgPgTNR ^&=O<>UVB>hQOiPK>^OL=T x)@]!PSRNHJIgTh1F TUP_`TDw]ghQTu@T`­`]!h<IgV OQ]PS`E@TIm`BA`yV?C6D TT`VPg]!PoR O@T(yVTUhQhiTNHJIgTV = x0+ Ker(A) ]!P@->/#SPgOiV`^ @gO<D TUP_`OQ]PR]7D D T

dim V = dim Ker(A) = n− rang(A).

Y O!ISP`BA`yV?C6D T eg]2D ]7\2CUPST ^lV]2I y]7I-=`ISPgT`]!h<IVOi]!P xÅh1F ]2=OL\OQPgT(y RTP!F T`V[jS^`R9=^O¦j4]2I-=hQTRU^!`\2>UPK>6=^h ;

Page 21: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

-: #9Q@9jk8 ) A !#"%$'&(H9XGOO q s;=>?K$,8Aj&(>@9! CA'>?3F"3U>? p c @"@'&(X"Np c b ∈Im(A)

b­P =T6D8^&=zHJIgT HJIgTRUTV V TR]!PK@OiV OQ]PF`6F Tfj-=OLDT~@g^PS` hF T`jS^!RT~@T`R]!hi]!PgPgT`Rm ; b­PF` ^OVw@T> 9H7ISThQTc` ]2IS`T`j_^RT

Im(A)T`V@Tw@gO<D TUP_`OQ]P

rang(A); b­P TUPt@->@TIgOiVMHJIgT` O

rang(A) < m hQ^ jK= ]7]S^&]gOQhQOV?>j4]2IK=VH7ISTb` ]OiVV@g^PS`

Im(A)T`V[P9IghQhiTTUV[hiT`BA `VBCD T`T=^lj-=T`?H7IST­V]2I y]7I-=`` ^PS`Y`]!h<IVOi]!P xR]!O<=#K\7I-=T

:7; W y ;dq^&= R]!PVB=TD`Orang(A) = m ]Pk@OiV HJIgT A

T`yV @T =^P-\ jghiTOiP hQT-`BA`yV?C6D T-^ @]PSR8V ]2I! y]2I-=`w@T`` ]hLIVOi]!PS` ; Y O @T8jghLIS` m = n hiT8PS]A!^&I @T AT`Vw@T @OLD TUPS` OQ]PFP9IghihQT TVhQT`[A`VBCDT^oIgPST`]!h<IVOi]!P

ISPgOLHJIgT ;*| TV V T` ]hLIV OQ]PKT`Vh1F OLD ^2\TN@TbjS^&=h^lVB=^PS`[b ]2=?D8^ VOi]!PDOQPJR!T6=`TO@T

A Pg]V?>UT A−1 F]_~aZ~x = A−1b.

. ]P_R`TIghiT`hiT`D8^ VB=ORT`qR^&=?=B>T`@T=^P-\ jghQTUOQP8`]!P!VqOQP9RT6=` O<]ghQT` ; b­P @gOV^2IS``O-HJIgTRT` VB=^PS`[b ]2=?D8^ VOi]!PS`` ]PV X[hjxzstow]_^@XZ ; T`D8^ V?= ORT`RU^&=?=?>UT`Pg]!P-OQP9RT=` O<]ShiT``]!P!V[^!` ` ]ROL>UT`Y^2ITfVB=^P_` b ]7=BD8^ VOi]!PS` V]a` xzstow]_^@XZ @gTRn ;

i 89jk+TG"#8 q <C(a>@o3"3U>?QG,I_UC"b c CG,aOp'?9O`?" # '?C"% c ;=>8 `A%!"b$& A3"mC>@IF"3U>?]O#"

rang(A) = rang ([A | b]) ,x :7;e_ y

>$#[A | b] "<Ud& ".K3a

m× (n+1)>^"%'C 'O>8F"b?C"oUAa>?U>?FX

b A ]_@'K`8d"%>8F"%fg. O>@

07 9C"o?O87 >?(?I" a'.a,U ;C9.aK3"%'&('C"0'7 C_?'''7 X& F".K.aON@"UdOG>@IF".>@AI"bG'"33_?=7 j F"b$&(N9CGO?9On;='K&(@" 8H"b"bfk a>?\I".>@ q Ssa>?& & '9 ;='K&(@" 8ea'.a,U 'l(A

X 0

Ker(A)

V

O

:2; U DN^2= OL>V?>hiOQP->^OL= T&% . ^PS`Rn Ker(A) = x ∈ Rn | Ax = 0 TV V = x ∈ Rn | Ax = b =

Ker(A) + x0

+-, 034=Mu>#Mu098 8J$2')$78 ) Y ]OiV

A ISPgTND8^ VB=ORT m× n; k ]PVB=T6=EHJIgT

kerAT = (ImA)⊥

ImAT = (kerA)⊥

.

d =B>RO`T=hiT`YT` jS^RUT`YR]!P!VTUPS^PVRUT`YTPS`TDw]ghQT` ;

Page 22: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

6

OIm(A)

b

:2;eW Y A`VBCDTlOiP_R]2D jS^V OL]ghQT % . ^P_`R

n b /∈ Im(A); . ]!PSR]!P PgTcj4T6IVVB=]2IKRT6=

x ∈ Rn V Th%HJIgT

b = Ax

Y ]!OVVD8^OQPV TPS^PVA =

1 2 0 10 1 1 01 2 0 1

.

2 IgThT`VhiTN=^P-\ @TA;. ]!PgPgT=hiT``]7IS`d T` jS^RUT`

ImA kerAT ImAT TVkerAT ;

87$7')$28k g »^ D8^ VB=ORTARUOwT@T`` ]2IS`E=TUjK=B>`TP!VT hiTj_^`` ^2\TN@Th^~]S^!`TR^Pg]!PgOLHJIgT ei1≤i≤4

9 hQ^]S^!`T A = ai1≤i≤4

;

A =

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

.

:2; d]7I-=Re_^P-\!T6=[hF ]7=?@-= Tu@T`hQOL\PgT`@TA ]PtD/IghVOijShiOQT A

jS^&=IgPgTwD8^VB=OQRUTP; ofPmPg]V^PV

ai hQ^hQO<\!PgTi@TX R]7D D TP!VjS^!` ` TVV ]P @T

a1

a2

a3

a4

9

a1

a3

a2

a4

o fj-=O<D T=[hQ^ D8^ V?= ORTA′ R]7=B=T` j_]!PK@g^P!VTTUV[hQ^ D8^ V?= ORT P

H7ISOG=B>^hQOQ` TRTV V T V?=^PS` b ]7=BD8^V OQ]P ;| ]2D D TUPVjS^!` ` TVV]!P @T

a1

a2

a3

a4

9

a1 − 2a3

a3 + 3a4

a2

a4 − a1

o fj-=O<D T=[hQ^ D8^ V?= ORTA′′ R]2=?=T` j_]!PK@g^PVTTV[h^ D8^ V?= ORT P

HJIgO =B>^hQOQ` TRUTV V T VB=^P_` b ]7=BD8^ VOi]!P ;T; . >UV T=BD OQPgT6=8hQT` D8^ V?= ORT`

P1 P2TV

P3HJIgOE=B>^hQOQ` TUPVhiT`VB=^PS`[b ]2=?D8^ V OQ]P_`l`BIgOLRª^PV T` `BI-= hQ^

D8^ V?= ORTT P TUP@T>(@TIgOL= T hF OQP9RT=` TM@gT T

;

T =

1 3 3 10 1 2 10 0 1 10 0 0 1

P1 →

1 3 3 00 1 2 00 0 1 00 0 0 1

P2 →

1 3 0 00 1 0 00 0 1 00 0 0 1

P3 →

1 0 0 00 1 0 00 0 1 00 0 0 1

Page 23: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

2P 8J$2')$78 L . ^PS`YV ]7IV[hF T6f T=RUOQRUT ]PPg]V T e1, e2

h^ ]S^` TRU^Pg]PgOHJIgTN@TR2 ;Y ]!OV u1, u2

ISPRU]2IgjShiTN@TNR!TR VT6I-=`hiOQP->^OL= TDTPVYOQPK@T>j_TPK@g^P!V`:%

u1 =

(

u11

u21

)

, u2 =

(

u12

u22

)

.

r PRTRV TI-=x@T

R2 `6F >R6= OiVx = x1e1 + x2e2 = x′

1u1 + x′2u2.b­PKOQhihLIS`VB=T6=^lhQT`=?>`BIghiV^V`E@T`EH7IST`V OQ]PS`^Jy 9 @QyY^XRTR­hQT`RTR VT6I-=`

u1 =

(

21

)

, u2 =

(

−11

)

.

:2; k ]PV?= T=HJIgT hF ]!PDj4T6IVV>R=O<=T(u1 u2) = M(e1 e2) ] M

T`yVEIgPSTMD8^VB=OQRUTMHJIgT hF ]!PDTfjghQOROiV T6=^ ;T; oPt@T>(@TIgOL= T

x1TUV

x2TP b ]!PSR VOi]!P@T

x′1

TUVx′

2

` ]2IS`b ]7=BD TOD8^ VB=OROQTUhQhiT ;P-; P9RT6=` T6=YhQ^ =TUh^ VOi]!P-j-=?>R6>@TPV T P ` ]OiV M−1 h1F OQPJR!T6=`TO@T M

;UK;| ^hRIghQT6= ‖ x′ ‖2 TPob ]PSRV OQ]P@T x1 x2

TUVV@T`E>h<>D TUPV`E@TM;

WT; X HJIgTUhQhiTRU]PK@gOVOi]!PK`BI-=u1TV

u2^&Z VBZt]!P

‖ x′ ‖2= λ2 ‖ x ‖2 .

^-; b­Pj_]`Tu11 = cos θ

TUVu12 = sin θ

;. >V T=BD OQPgT6=hQT`D8^VB=OQRUT`M+

TVM−

VTUhQhiT`EH7IST‖ x′ ‖2=‖ x ‖2 .

_9;2 IgTUh`[` ]PVYhiT`ERTR V TI-=`OQP9R ^&=OQ^P!V`jS^&=M+

TVM−

-;VY ]OiVxhQTRTR V TI-= @TKRU] ]2=z@]PgPK>UT`

x1 = 1TUV

x2 = 2; k ]!PVB=T6= H7IST

M+x`F ]2]VOiTP!V jS^2=/IgPST

= ]V^V OQ]P @TθTVEH7IST

M−x`F ]7]V OQTUPVYjS^&=IgPgT `BAJD >UVB=OiT­jS^&==^jgj4]2= V9lhQ^/@T= ]!OVTM@!F ^P-\hQT

θ/2; b­PPg]VT6=^

M+ = RθTV

M− = Sθ/2;| ^hRIghQT6=

R−θTUV

(Sθ/2)2 ; oP@T>(@TIgOL= T R−1

θ

TUVS−1

θ/2

;

Page 24: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

U 6

Page 25: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

#1 ) & ( !

JG,.+ : 0 0KJ*2BU@0Y ]!OV[hQT`BA `VBCD TO@gT P >(HJIS^ VOi]!PS` 9 P OQPSRU]PgP9IgT`

2x− 3y = 3x T;L: y

4x− 5y + z = 7x T; y

2x− y − 3z = 5x T; P y

»^uD >Veg]T@TO@GF >hiOLD OiP_^ V OQ]P @TOv ^2IS``xÅ]2I D >UV eg]T@TM@TIDjgOLR]VzyR]!PS`O`V T9wIV OQhQOQ` T6=hQ^cj-=T6D O<C= T>HJIS^V OQ]Pj4]2IK=RU^hQR6IghQT6=xTUP~b ]!PSR VOi]!P @T`^&IgVB=T`SR ^2= O^&]ghQT`j-IgO`4@T= TD jghQ^!RT=fRTUVV TR ^&=OQ^2]ghiT@g^PS`fhiT`S>(H7I_^ V OQ]P_``BIgOLR ^PV T` ;-| TV V TO>UhQO<D OQPS^ VOi]!P-` Tj4]2I-=`BIgOV^XRTR

y@g^PS`hiT`PS]2I-R!TUhQhiT` >(H7I_^ V OQ]P_`x ` ^2ITbh^cj-=T6D OLC6=T(y IS` Z

HJI!F 9 hF ]7]V TP!VOi]!P~@GF ISPgTV>HJIS^V OQ]P&9OIgPgT[` T6IShiTOQPSR]!PgP9IgT ; b­P =T6D ]!P!VT[^hQ]2=`TUP =T6D jgh^ U^PVhiT` R ^&=OQ^2]ghiT`R^hRIghL>UT`E@g^P_`YhiT`>HJIS^ VOi]!PS`^XA!^PVY` T6=?R O 9 hF >UhQO<D OQPS^ VOi]!P %` OL=]PS`

x@T hF >(HJIS^ VOi]!P x T;L: y1%

x =3

2(1 + y)

x T; U yZ[T6D jgh^ ]!PS`

x@g^PS`YhiT`E@T6I-fo@T= PgOLC6=T`>(H7I_^ V OQ]P_`Nx T; y Zzx T; P yfjS^&=[` ]PKTfj-=T`` Oi]!P x -; U y

y + z = 1x T;eW y

2y − 3z = 2.x T; ^ y

` OL=]PS`y@gThF >HJIS^ VOi]!P x T;eW y@TRUTPg]2I-R!T^2I-`BA `VBCD T6%

y = 1− z.x T; _ y

Z[T6D jgh^ ]!PS`y@g^PS`YhF >(H7I_^ V OQ]P x -; ^ y%

−5z = 0.x T; y

»^jge_^` TM@GF >UhQOLDOQPS^V OQ]PDT`yVV T=BD OQP->UT ; b­PDT ¦TRVBIgT^hi]7=`fhQ^`[IK]S`yVOV?IV OQ]P-TUP`TPS`OiP9RT=` T@gT`R ^&=OQ^2]ghiT`\7=!RT ^&ITf >HJIS^ VOi]!PS`Nx -; y x T; _ yTVwx T; U y%(2.8) ⇒ z = 0

(2.7) ⇒ y = 1

(2.4) ⇒ x = 3b­P=T6D8^&=zHJIgTNH7I F 9 IgPSTN>V^j_Tu@g]PgP->T hF >UhQO<D OQPS^ VOi]!P@!F IgPSTNRª^2= O^&]ShiT j4T6IV­` TObÅ^OL=Tu@g^PS`YP F OLD j_]7=VTHJIgThihQTO>HJIS^V OQ]P 98RU]PK@OiV OQ]P ]gOiTP ` -= HJIgTRTUVV TN>(HJIS^ VOi]!PKRU]PV OQTUPgPSTh^ R ^&=OQ^2]ghQT­TPHJIgT`yVOi]!P ;7W

Page 26: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

&^

J,'J E8F;=HKJL098 ;47>@<A:ARBU@<A>V47098b­P R]!PS`yV^ VT6=^l^2IKj_^&=^&\2=^jSegT`BIgOLR ^PVHJIgThQ^ jSeS^` Tu@GF >UhQO<D OQPS^ VOi]!PRU]PS` O`yVT 9V?=^PS`[b ]2=?DT=YhiTc`BA` ZVBCD T]2=O<\!OiP_^hGTUPISPm`[A`VBC6D T V?= O^P-\7Igh^OL= T ;-r Pm`BA `VBCD T VB=OQ^P-\2Igh^OL=T­T`yVISPm`[A`VBC6D Tu@]!PV[hQ^ D8^ V?= ORTT`VVB=O^P-\7IghQ^O<=T ;r PSToD8^VB=OQRUT bfXV]_iv` xzstopik]aX[Z Vsech@XV]_Z.stX[Z x=T` j4TR VO<R!T6D TUPV blXV]_ik` xzstopiv]aXZ ]a`@h.XV]_Z.stX[Z yT`V IgPST D8^ V?= ORTRU^2=B=?>UTO@]!P!V[hQT`>h<>D TUPV`` ]2IS`Yh^ @O^&\]!PS^hQT x=T` j ; ^&I@T` `BIS`zy` ]PVP9Igh` ;r P`BA`yV?C6D TVB=O^P-\7IghQ^O<=T `[ISjQ>= OQT6I-=-`T=?>` ]2IVj_^&=D`BI-]S`V OiVBIVOi]!P^&=?= OLC6=T x ]!P `BIgjgj4]!` T6=^´V ]7IS`8hQT`

>UhL>6D TP!V`V@O^&\!]PS^2ITfaii

Pg]!PPJIShQ`zy ;a11x1 + a12x2 + · · ·+ a1nxn = b1

a2nx2 + · · ·+ a2nxn = b2

· · · · · ·annxn = bnb­P R]7D D TPSRTN@GF ^&]4]2=z@Dj_^&=R^hRIghQT6=

xn@g^PS`hF >(HJIS^ VOi]!P

n; d IgOQ`9 h1F ^O@Tu@T

xn ]Pj4T6IV­R^hRIghQT6=xn−1

@g^PS`YhF >(H7I_^ V OQ]Pn− 1

TV^OiPS` O!@T`BIgOiV T IS`?HJI!F 9x1;Q| TuHJIgO#@]PgPSThF ^hL\]2=OiV e-D T `[IgOLR ^PV ;

xn =bn

ann,

x T; c y

xk =1

akk

bk −n∑

j=k+1

akjxj

, k = n− 1, . . . , 1.x -;<:(d y

b­Pn=T6D8^2=?HJIgT H7ISTlhQTR^hRIgh @Txk

R]V Tn − k

_]jS`TVOIgPgT~@OLR OQ` OQ]P ; T R]VV ]V^h @Th1F ^hL\]7= OiV eKDTT`V@]!PSRO@T1 + 2 + · · ·+ n− 1 =

n(n− 1)

2,

` ]OiVwx ]PKPgTN\^&=z@TMHJIgThQT`VT6=?D T`E@TjghLIS`YeS^2IV@T\2=?>(yn

2

2

+p8Jn

%'87!'1p'1Q%pz. ^PS`YhQTRU^!`E@GF ISP`BA`yV?C6D TVB=OQ^P-\2IShQ^O<=T­OQPTb>= OQT6I-= ]PKT ¦TRVBIgTu@gT`[`BI-]S`V OiVBIVOi]!PS`E@OL= TR VT`TUVYhF ^hL\]&Z=OVe-D TT`V

x1 =b1

a11,

x T;L:2: y

xk =1

akk

bk −k−1∑

j=1

akjxj

, k = 2, . . . , n.x -;<: y

TRU]VV@T hF ^h<\!]2=OVe-D TM@T`BI-]S`V OiVBIVOi]!P x T;L:2: y Z x -;<: yqT`yVhQTOD6,D TOHJIgTRUTUhLIgO @T`BI-]S`V OiVBIVOi]!P-OQP9RT6=` Tx -; c y Z x -;<:(d y ;

J, 6Y; B?GC50CB0 <ARB88?AR CBR 2B> A?); 2O?AR 4U@098 8EG8;=HKJL098&U.>:B69<A>V47098

k ]!P!V?= ]!PS`S@GF ^&]4]2=z@ HJIgT[h^V TReSPgOLHJIgT@GF >hiOLD OiP_^ V OQ]P @->R=OiV T^2I T;L: R]7=B=T` j4]PK@69NIgPgT]j*>6=^ V OQ]P~@TjgOLR]V^&\!T ;

Page 27: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

1 Qq @ :0s J_ hQOLDOQPgT=

x@T-h1F >HJIS^V OQ]P,x -; yN=T6R OQTUPV69tbÅ^OL= Th^ ` ]2D D T @TKhF >(H7I_^ V OQ]P,x T;L: yuDwIghiV OQjghQO<>T-jS^2=uZ ^XR!TRx T; y ; . T D0,6D T >hiOLD OiPST6=

x@T x T; P yu= TR OiTP!V69bÅ^OL= TDhQ^ `]7D D T @gTDhF >(H7I_^ V OQ]P,x T;L: yuDwIghiV OQjghQO<>Tj_^&=Z : ^XRTRhF >HJIS^ VOi]!P x T; P y ; b­P^ @]PSRT4TR VBIK>MISP-jgOLR]V^2\TO@T hQ^lj-=T6D O<C= T R]!hi]!PgPgT xÅRUTUhQhiTN@T

xyTP

IgV OQhiO` ^PVYh^ j-=T6D OLC6=T­hQO<\!PgTR]7D DT hQO<\!PgTN@TIjgOLR]V ;i 89jk+TG"#8tgF) a r*8 I"bG'"3U>? c 7 GUG& C"f8DkBaO>@>@FCm8?!NU`l:CH!e;C3_?>8"f8>?3" ?".O

X>@jC Oe".?*Sg3>?p& "3U>?!DGUG& C"I?9OD\@a"S8G#D# U & ".K3a>@C"o,a,"SGd'B;o'$'

'@a>?\A&(&B^?O UAEF 8D'7 G'99&A9*F".>@ q9p6"b>F"n;oK'`^?' s c >@ >^F".'C"N]%!"b$& `"3pU?Fl*U?9D_@@aNUO

;CI_@>8"IN#NkBU@lF>@*U T`YV?=^PS` b ]7=BD8^V OQ]PS``[I_RURT` ` OLRT`Y` ]PVV@T>R6= OiV T`RO<Z @gT``]7IS`j_]7I-=h1F TfTDjShiTu@TI T;L:2;

V?>6=^ VOi]!P : %

1−2 1−1 1

(2) −3 0 34 −5 1 72 −1 −3 5

=

2 −3 0 30 1 1 10 2 −3 2

V?>6=^ VOi]!P %

11−2 1

2 −3 0 30 (1) 1 10 2 −3 2

=

2 −3 0 30 1 1 10 0 −5 0

. ^PS`hQTR^`4\2>P->6=^h ]!PD^/@g]PSR9hF >UV^j4T k@T­h1F ^hL\]7= OiV e-D TISPgTD8^VB=OQRUT

A(k) @gT nhQO<\!PgT`TV

n + 1RU]hQ]PgPST`uxÅhQ^~@T= PSO<C= TRU]hQ]PgPgTN=TUjK=B>`TP!VThiTl`TR]PQ@ D TDw]-=Tu@TIm`BA `VBCD TVB=^PS`[b ]2=?D >(y@]!PV[hiT`k − 1jK= TDOLC6=T`YRU]hQ]PgPgT``]!PVV?= O^PK\2Igh^OL= T``BIgj*>6=OiTI-=T`NxÅRb ; #K\7I-= T -;<: y ;

a(k)kk

[

A(k)|b(k)]

=

T;L: V^j4Tk@Th^/D >UV eg]T@Tu@Tuv ^2IS` `

b­P D TV @]PSR&9>6=] ReS^2HJIgT~>h<>DTPVw@Th^ R]hQ]PSPgTk`]7IS` hiTDjgO<R!]V

a(k)kk

x `[IgjSj_]`[>8Pg]!PFPJIShy TP=T6D jgh^ ^PVYhQ^ hQO<\!PgT

ix j4]2IK=

i = k + 1, . . . , nyjS^&=

hQO<\!PgTi ← hQO<\!PgT

i +−a

(k)ik

a(k)kk

× hiOL\PST k.

Page 28: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

&

SF >UhL>6D TP!Va(k)ij

j4]2I-=i = k + 1, . . . , n

TUVj = k + 1, . . . , n + 1

@T6R OQTUPV

a(k+1)ij = a

(k)ij −

a(k)ik

a(k)kk

a(k)kj

^ D O`T 9 y]7I-= @T h^FhiOL\PgTi= T(H7ISOiT=V @]!PSR

n − k + 1_]jS` TUV : @O<R O`OQ]P ; . ]!PSRh1F >V^j4T

kRU]VT

(n− k)(n− k + 1)S]!jS`TV

n− k@OLR OQ` Oi]!PS` ; b­P-TP@->@TIgOiV[hQTR]gVV]V^h @T hF ^h<\!]2=OVe-D TN@Tuv ^&IS`` %

n(n− 1) + (n− 1)(n− 2) + · · ·+ 2 · 1 = n2 + (n− 1)2 + · · ·+ 22 + 12 − (n + n− 1 + · · ·+ 2 + 1)

=1

6n(n + 1)(2n + 1)− 1

2n(n + 1)

S]!jS`TV

1

2n(n + 1)

@OLR OQ` OQ]PS` ;b­P@OL=^ HJIgT hQ^ RU]2D jghQTfOV?>u@T hQ^ D >Veg]T@Tu@Twv ^2IS` `YT`yV@gT 1

3n

3%+%p

SF ^hL\]7= OiV eKDT @GF >UhQO<D OQPS^ VOi]!P @T v ^&IS``T`yVj-=?>` TUPV?> @S^PS` hF ^h<\!]2=OVe-D T :2; T` TRU]PK@ D T6D/]-= T8T`yV`V ]R >O@g^P_`h^ @T6=PgOLC6=TRU]hQ]PgPgTO@TA; F >hiOLD OiP_^ V OQ]PDVB=^PS`[b ]2=?D T­h^/D ^VB=OQRUT

ATUPoIgPgTOD8^ V?= ORT­VB=O^PTZ

\2IShQ^O<=T`[Igj*>6=OQT6I-=T~x h.bjireZz~ y ; T`BA`yV?C6D TVB=OQ^P-\2IShQ^O<=T`BIgj*>6=OiTI-=T`VTUPS`BIgOiV TN=?>` ]hLIDjS^2=[`[IK]S`yVOV?IV OQ]POQPJR!T6=`T x h.bdiecZ ~ y ;' 0 QQ'#jk8 k>UV eg]T@Tu@gTwv ^&IS``z~ oy]a\]a`givbf]_mv` *"

k = 1, . . . , n− 1 +-')8 Q"#i = k + 1, . . . , n +K'8 *"

j = k + 1, . . . , n + 1 +K'8aij ← aij −

aik

akkakj

') Q"# '1 *"

') *"# ~h .mvoystbl]_mk` s (Vbj^.\GZAbfXV]_iv` xkstopik]aX[Zxn ←

an,n+1

ann *" k = n− 1, . . . , 1 +-')8

xk ←1

akk

ak,n+1 −n∑

j=k+1

akjxj

') *"#

vfuNuSt F|zn~ 0xmn 0xD~ nFf SY OhiTDjgO<R!]VcT`yVP9IghxÅTUPIj-=^ VOLHJIgT ]!P >6R OVT6=^m^&I_` ` OfhQT`cjgO<R!]V`w@T R ^hiTI-=l^2]S`]!h<IST8VB=]j j4TVOVT R b ;ReS^jgOV?= T P y ]P hQTt=T6D jgh^RT TUP T ¦TRVBIS^PV IgPgT j_T=BD/IV^V OQ]P^XRTRIgP >UhL>6D TP!VKPg]P P9Igh­jS^&=?D OhQT`>UhL>6D TP!V`[` ]2IS`YhLIgOGTV ª]7I 98`^/@-= ]!OVT ; b­P@O`yVOiP-\7IgTO\2>P->6=^hQT6D TUPVV@TITf-`VB=^ VB>\OQT`1%

'87*K(+ *8Q+-0 b­Pj4T6=?DwIVTfhQO<\!PgT`TVR]hQ]PSPgT`j_]7I-=»Reg]!OQ` OL=hQTfjghLIS`!\7=^PK@O>UhL>6D TP!V TUPuR ^hiTI-=»^2]S`]!h<IST@g^PS`Yh^ `]7IS`[ZD8^ V?= ORTTUP]S^!`9 @T= ]!OVT~xR]!O<= OL\ ; -;<: y%

max∣

∣a(k)ij

∣ ; i = k, . . . , n; j = k, . . . , n

Page 29: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

1 Qq @ :0s 2c '*7QQ+ *8 ++TX'18J0 b­PPgTj4T6=?DwIVTMHJIgT­hiT`hQOL\PgT`` ]2IS`fhiTjgOLR]VTUPKReg]O` OQ`` ^PVfhiT­jghLIS`\2=^PK@ >h<>DTPVTUPR ^hQT6I-=[^2]S` ]hLIgT

max∣

∣a(k)ik

∣ ; i = k, . . . , n

»^c`VB=^ VB>\OQTV@TIjgOLR]V^2\TjS^2=VOiThST`Vh^jghLIS`4IV OQhQOQ`B>UTRU^&=fhQ^jSh<IS`4>RU]Pg]7D OLHJIgTwxÅhQ^u= TRegT6=RegTV@TIjSh<IS`\7=^PK@ >UhL>6D TUPVl`BI-=uISPgTDhQOQ`V T @T

pPg]7Dw]-=T`cR]gV T

pR]7D jS^&=^O`]!PS` P9I-D >6=OLHJIgT` RT HJIgO@]PgPST n3/3RU]2D jS^2=^OQ` ]PS`j4]2IK=fhQT[jgOLR]V^&\!TYV ]V^h y ; SF ^hL\]2=OiV e-D TV@GF >UhQO<D OQPS^ VOi]!P @Tv ^2IS` `^XRTR= TRegT6=RegTE@TIjgOLR]Vj_^&= V OQTUhT`yV[j-=?>` TUPV?>O@S^PS`hF ^hL\]2=OiV e-D T T; T`TR]!PK@oDTDw]-=T T`V­`V ]R >N@g^PS`h^~@T6=PgOLC6=T RU]hQ]PgPSTO@gT

A;

' 0**K')%jk8g k>Veg]T@TN@Twv ^&IS``^XRTRjSO<R!]VjS^&= V OQTUhz~ ow]a\S]a`ivbf]_mv` *"#

k = 1, . . . , n− 1 +K'8ZrqcZ@XqcZ sSe]vmvbcp ← |akk |, ip ← k *"

i = k + 1, . . . , n +-'8 ' |aik| > cp' 01Kp

cp ← |aik |, ip ← i '1 '

') *" W9Z@XV\Sstbdivbf]_mv`!'

ip 6= k' 01KpdT6=?DwIgV T6=hQT`YhQOL\PgT`

ipTV

k@ThQ^ D8^ V?= ORT

A '),!'WY]mkbji xZ *"

i = k + 1, . . . , n +-'8 Q"#j = k + 1, . . . , n + 1 +-'8

aij ← aij −aik

akkakj

'1 Q"# ') *"

') Q"# ~ h @mkowstbf]_mv` s (Vbd^@\GZ)bfXV]_iv` xksto iv]aX[Zxn ←

an,n+1

ann *"#k = n− 1, . . . , 1 +K'8

xk ←1

akk

ak,n+1 −n∑

j=k+1

akjxj

') Q"#

vNufu3v T|Q8~S n p~ o \ C|S\D D F| .D |!~SC '*7QQ+ *8(K(+-0

Y O 9 hF OiVB>=^V OQ]P k@gThQ^ D >Veg]T@Tu@Tuv ^&IS``^XRTRjgO<R!]VYV ]V^h

max∣

∣a(k)ij

∣ ; i = k, . . . , n; j = k, . . . , n

= 0

Page 30: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

P2d

x)@g]PSRV ]7IS`hiT`4>h<>D TUPV`@ThQ^`]7IS`[ZD8^ V?= ORT­`]!P!VP9IghQ`zy ]!PDj4T6IV[^ ~=?D T6=4HJIgThQT=^P-\ @ThQ^wD ^VB=OQRUT AT`V>\!^h9k − 1

;Y F OQh TfOQ`V TwIgPt>UhL>6D TP!Vb(k)i k ≤ i ≤ n @TIm` TR]!PK@D T6D/]-= T/@O>6=TUPVO@T6>= ] ^hQ]2=`hQT`[A`VBC6D T P!F ^ jS^`E@gT ` ]hLIVOi]!P ;

'87*K(+ *8 +%+-((')890Y O

max∣

∣a(k)ik

∣ ; i = k, . . . , n

= 0]PDj_TIVY` T6IShiTDTPVY^ =?D T6=H7ISTh^cR]!hi]!PgPgTkT`VhQOiP->^OL= TD TUPV@T>j_TPK@g^PVT@T`

k− 1j-=T6D OLC6=T` ;9| TUh^OLDjShiOHJIgT HJIgT

rang(A) < nTVuHJIgT hiT`BA`yV?C6D TP!F ^jK= ]7]S^&]ghQT6D TP!V jS^!`N@T8`]!h<IVOi]!P ;!` ]2IVTb ]O` ]P´PgTj4T6IVTP,UVB=T` K=VH7IST `BI-=hQT V T`yV@TIjgOLR]VYV ]V^hGP9Igh ;

i 89jk+-XG"%8ngFg 0B&(G@".rC>(8 Ke@"N "%"! ;C3_?>8""b>"I?Cn;F;o'9K'C"8>?\a a>@& &(Hk&('9@"."%GKl: ;>da'.a,UNU ?lj7 XD& F".K.aO

J, <HM ;=09R5478 C NRB:B0 JL<);4I>#Mu0 :B?A:'8=>@: -R5U@>.H34I02 IS^PK@]P^YjghLIS` OiTI-=``[A`VBC6D T`hQOiP->^OL= T` 9E=?>` ]2IK@T=Tq^XRTR»h^ED0,6D TSD8^VB=OQRUTqTV#@T`` TRU]PK@g`!DTDw]-=T`@O*>= TPV` ]Pl^OQPVB>6= ,VhQ]2=`%@gTh^­j-= TD O<C= T=B>`]!h<IgV OQ]P 9M\!^2=?@gT6=hQT`qRU] T ROQTUPV`@T`jgOLR]V^&\!T` `[ISRRT` ` O bÅ`TUP D >6D ]OL= T ;T| ThQ^lR]7=B=T` j_]!PK@&9u\^&=z@T6=hQ^ubÅ^RV ]2=O` ^V OQ]P

LU@gT­h^uD8^ V?= ORT ; ofPDT ¦TV ReS^2HJIgTjgO<R!]V^&\Tj4T6IV ,UVB=To=TUjK=B>`TP!V?>-j_^&= IgPgTD8^ V?= ORTo>UhL>6D TP!V^OL= THJIgO[PgT@OC6=T@gTKhF OL@TPV OiVB>tHJIgTjS^&=~ISPgT `]7IS` ZR]!hi]!PgPgT ;ZTj-= TPg]P_`8hiT jgO<R!]V^&\Tt@gTmhQ^kD ^VB=OQRUT

A(k) 9 h1F >V^j4T k;Y ]OiV

ηkhQTtR!TR VT6I-= @T

Rn−k @]PVDhQT`R]7D j_]` ^P!VT`Y` ]PVηik = a

(k)ik /a

(k)kk

; b­P ^ @]PSR xÅTUP `[ISjgj_]` ^PVa(k)kk 6= 0

yA(k+1) = EkA(k)

]Ek

T`V[h^/D8^VB=OQRUTM>h<>D TUPV^O<=T`[IgOLR ^PV T

Ek =

1 ; ; ;−ηk 1

b­P ^ @]PSR ^j-=?C` n− 1jgOLR]V^2\T`

A(n−1) = En−1En−2 · · ·E1A.x T;L:P y

]V ]!PS`UhQ^ D8^ V?= ORTVB=O^P-\7IghQ^O<=T `[Igj*>6=OQT6I-=T

A(n−1) TVE=?>6>R=O<R!]PS`h^/=TUh^ VOi]!P x T;L:P yA = E−1

1 · · ·E−1n−1Ub­P~R2>= O$#STHJIgThQ^ND8^VB=OQRUTOiP9RT=` T

E−1k

^h^ND0,6D TVb ]7=BD TVHJIgTEk

^XRTRYhQT` >UhL>6D TUPV` @T[h^` ]2IS`[ZÀR]hQ]PSPgTkReS^P-\2>` @T`OL\PSTTVHJIgThQT`j-=]9@-IgOV`

E−1k E−1

k+1

`6F T ¦TRVBIgTPVY`^PS`RU^hQR6Igh_TPD^RR]!hQ^P!VhiT` R!TR VT6I-=`ηkTV

ηk+1@g^PS`hiT`fR]!hi]!PgPgT`

kTV

k+1; b­P8j_TIV4@]!PSR>R6= OL= T

A = LU]

LT`V IgPSTVD8^ VB=ORTYVB=O^P-\7IghQ^O<=TOQPTb>6=OiTI-=TO@g]PV[hiT`>UhL>6D TUPV`V@O^&\]!PS^&I-f8` ]PVE>6\^&ITf 9 : TUV[hiT`>UhL>6D TUPV``]7IS`YhQ^ @O^&\!]PS^hiT ` ]PV

lij = ηij .b]S` T6=?R]!PS`NHJIgT8hiT`N>UhL>6D TP!V`Pg]!P @OQ^2\]!PS^&ITf @TLj4T6I-R!TUPV ,V?= T~=^P-\7>`N@OL=TR VT6D TUPV 9h^jgh^RUT @T`

>UhL>6D TP!V`4@gTARU]2=?= T`j4]PK@S^PV` ; ^OD8^ V?= ORT

AT`V4@]!PSRE=TR]7I-RT=VTYjS^&=` ^ObÅ^!R V]2=OQ`^ VOi]!P

LUTVhQTRU]V

@T`V ]R ^&\!T­T`yV[TPn2 ;

Page 31: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

j7 PK:vM= RUT 9FRUTVVTbÅ^RV ]2=O` ^V OQ]Px1HJIgO[PgTmRU]VTt@]PSRKjS^!` jghLIS`RegT= H7ISTKh^´VB=O^P-\7IghQ^2= O`^ V OQ]P*y V ]2IgVPS]2I-R!T^&I`BA`yV?C6D ThiOQP->^O<=T

Ax = b′j4T6IgV,UVB=T=?>` ]hLI8jS^&=h^u=?>` ]hLIV OQ]P @gTM@TITf`[A`VBCDT`V?= O^PK\2Igh^OL= T`

x1@]P_R­TPO(n2)

S]jS`zy ; oPT ¦TV j_]7I-=E=B>`]7IK@T=TLUx = b′]!P =?>` ]2IgVE@!F ^2]_]7=?@Ly = b′jKIgOQ`Ux = y.

' 0**K')%jk8o ^RV ]7= O` ^V OQ]PLU

σ(i) = i i = 1, . . . , nx OQPgOVOQ^hiO`^ V OQ]P@TIR!TR VT6I-=E@T`j4T6=?DwIV^ V OQ]P_`?y

*"#k = 1, . . . , n− 1 +K'8

ZrqcZ@XqcZ sSe]vmvbcp ← |akk |, ip ← k *"

i = k + 1, . . . , n +-'8 ' |aik| > cp' 01Kp

cp ← |aik |, ip ← i '1 '

') *" W9Z@XV\Sstbdivbf]_mv`!'

ip 6= k' 01KpdT6=?DwIgV T6=hQT`YhQOL\PgT`

ipTV

k@ThQ^ D8^ V?= ORT

Aσ(k) = ip σ(ip) = k

'),!'WY]mkbji xZ *" i = k + 1, . . . , n +-'8

Z@\eow]".i.x Z Z-opi qrmkopmv``Z k ecivXAo Z -qrm Z q@]_Z.`cb ηik

aik ←aik

akk m ] uqrikbl]_mk` Z )ow]yxz`Z @st] ` mk`cbeci(-Z.`qrmvXZh.bjhAow]yxz`Z e]vmvb Q"#j = k + 1, . . . , n + 1 +-'8

aij ← aij − aikakj

'1 Q"# ') *"

') Q"#2 IS^PK@^&ISR6IgPcjgOLR]VqPJIShP!F T`yV =TUPSRU]PVB=?> A j_TIVf` TD TUVVB=T`]7IS` hQ^b ]2=?D T

LUTUVqRUTV V TbÅ^RV ]2=O` ^V OQ]PT`yVOIgPgOHJIgT ; oP´T ¦TV `6F OQh T6fOQ`V T @T6I-fbÅ^R V]2=OQ`^ VOi]!PS` L1U1

TUVL2U2

@TA ]!P ^-^hi]7=` L1U1 = L2U2

;| TMHJIgO_OLD jghQOLHJIgTMH7IST

L−12 L1 = U2U

−11

TVhQTjK= ]T@TIgOiV@gT@TITf~D ^VB=OQRUT`fOiPTb>= OQT6IK= T`x=T` j ; `[Igj*>6=OQT6I-=T`zy>UV^P!VMIgPgT D8^ VB=ORTV?= O^PK\2Igh^OL= TcOiPTb>= OQT6IK= Tox=T` j ; `BIgj*>6=OiTI-=T(y RT`j-= ]T@TISOV`­` ]PV­PK>RT` `^OL=T6D TUPVIgPSTD8^VB=OQRUTO@gOQ^2\]P_^hQT ;K| F T`yVhF O@TP!VOV?>cRU^2=(l1)ii = (l2)ii = 1

j_]7I-=YV ]7IVi;

. ^PS`hiT8RU^!`O@GF ISPgT`yV?=^VB>\OQT @T jgOLR]VjS^&= V OQTUh `O PkT`VhQ^oD8^ V?= ORT~@T j4T6=?DwIV^ V OQ]P @T` hiOL\PgT`:9hF OV?>6=^ V OQ]P

k ]!PKj4T6IgVV>R=OL= TA(k+1) = EkPkA(k).ofP bÅ^OV hQT`E@O*>= TPV T`[j_T=BD/IV^V OQ]PS`Yj4T6I-R!TUPV,V?= TO=?>`BI-D >UT`E@S^PS`Yh^ D ^VB=OQRUTP = Pn−1Pn−2 · · ·P1TUV[]PK]2]VOiTPV[hQ^ @T>R]2D j4]!` OVOi]!P \7>UP->=^hiT `BIgO<R ^P!VT ;

Page 32: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

P

%9K?9jk8tg1 6>6"%>8F"%& ".K3aA

X>?eS9l*$'O67 >?#:n c 98 8k"b X=& F".K.aOn8C;='K&(F"b".>@

P c X& "3p3a ".Kkl*U?9O9Fg3Gp@OL

"bAlii = 1 c ;>H"%>8F"

i c @"DX& "3p3a ".Kkl*U?9O;Gp@

U c "%'9UOPA = LU.

g ­b­P^En−1Pn−1 · · ·E1P1A = U

; b­PD]!PVB=T ^hQ]2=` HJIgTmh^kD8^ V?= ORTL =

P (En−1Pn−1 · · ·E1P1)−1 T`V4]gOQTUP8V?= O^PK\2Igh^OL= TOQPTb>= OQT6I-=T ; b­P =T6D8^&=zHJIgTVHJIgT@S^PS`RTRU^` |lij | ≤ 1

;SF ^hL\]7= OiV eKDT P D]!PVB=T-hQT` @O*>= TPV T` >UV^j_T` @gTKh^ bÅ^RV ]2=O` ^V OQ]P

LU^XRTR =TRegT=RegTo@TIHjgO<R!]VjS^2=VOiTh ; oPKTUPV?=B>T­]!PK^lhQ^ D8^ V?= ORT

ATVhQTMR!TRV T6IK=@gT`j_T=BD/IV^V OQ]PS`

σ; ofPK` ]2= V OQT hQT` A

R]PVOiTP!VYhQT`bÅ^RV TI-=`

LTUV

U@T h^/D8^VB=OQRUTTUV

σhQT`Yj4T6=?DwIgV^ VOi]!PS`E@T hQO<\!PgT`>RTUPV?IgTUhQhiT` ;KY O

σi = j^hQ]2=`hiT`YhiOL\PST`

iTV

j]PVV>V?>j4T6=?DwIV?>UT` ; T`j4T6=?DwIgV^ VOi]!PS`@g]OLRTUPV,UVB=TN= Tj_T=R6IVB>T`[`BI-=hQT` TR]!PK@D T6D/]-= ThQ]2=`V@Th^ =B>`]!h<IVOi]!P @T

Ly = b qmk`@~ ^hL\]7= OiV eKDT UK;' 0 QQ'#jk8 Y IK]S`yVOV?IV OQ]P_`V@OL= TR V T` OiP9R!T6=`T`z~ s & Vbf]ablstbf]_mv`) ]aXZq.bdZ Ly = bx1 ← bσ1 *"

k = 2, . . . , n +-'8

xk ← bσk−

k−1∑

j=1

akjxj

') *"# ~ s & Vbf]ablstbf]_mv`)]a` vZ.X"@Z Ux = y

xn ←xn

ann *" k = n− 1, . . . , 1 +-')8

xk ←1

akk

xk −n∑

j=k+1

akjxj

') *"#

vfu uSt T|yx |=F~p\ S~So!35+T')$289p~p)!jkJ6')G"%89p

. ^PS`8RUTKR^` Uj4T6IV`F >R=O<=T

U = DLT ]DT`Vh^ D ^VB=OQRUTo@gOQ^2\]P_^hQT-R]!PV TUP_^PVhQT` jSO<R!]V``BISRURUT``O<bÅ` ; b­P ^ @]!PSRhQ^/bÅ^RV ]7= O` ^V OQ]P

A = LDLT ; ^ R]7D jghiT6fOV?>O@gTh1F ^hL\]7= OiV e-D TT`V^hQ]2=`@T n3/6S]!jS`uxÅRb ; TfT=RUOQRUT(y ;35+T')$289p +-% #89p

| T` ]PVV@T`ED8^VB=OQRUT``BAJD >UVB=OLHJIgT`V TUhQhQT`VHJIgTaij = 0

j_]7I-= |i− j| > pxpT`V[hQ^ h^&=?\TI-=E@TN]S^PQ@T

@T h^ D8^ V?= ORT p < ny ;!| T`MD ^VB=OQRUT`­OiPVT6=?ROQTUPSPgTUPVR]2IK=^2D D TP!VO@g^P_`­h^o@OQ`R=?>VOQ`^ VOi]!Pn@GF >HJIS^V OQ]PS`@O*>= TPV OQTUhQhiT` ;# hT`Vl^hQ]2=`bÅ^RUOihQT @T~D ]PV?= T=uH7ISThiT`ObÅ^RV TI-=`

LU= T`j4TRV TUPVh^]S^PQ@T ; b­P ^ ^hi]7=`OQP!V?>6= ,VV@T `yV]R T6=h^wD8^ V?= ORT x TUV[`T` bÅ^!R VT6I-=`

LUy`]7IS`hQ^wb ]2=?D TM@T­V^2]ghQT^&I 9

nhiOL\PST`TV

pR]hQ]PSPgT`TV[h^ R]7DjShiT6f OiVB>T`yVTP

np2/2S]!jS` ;

35+T')$289p~p)!jkJ6')G"%89p/ % 4%%')89p + *p')'87*89p| T` D8^VB=OQRUT` `T= ]!PV >UVBIK@OL>UT`8^&I ReS^jSOV?= T ^-; ohQhQT`8j4]!``[C(@TUPV IgPgTbÅ^!R V ]7= O`^ V OQ]P IgPgOH7IST

LDLT^XRTR~@T`cjgOLR]V`c`BISRURUT``O<bÅ`c`VB=OR V TD TUPVcj4]!` OVO bÅ` ; »^obÅ^RV ]2=O` ^V OQ]PFj_TIV`F T ¦TR V?IgT6= @OL= TR V TD TUPV`^PS`

Page 33: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

1 P7PjSO<R!]V^2\TjS^&=OL@TPV O$#_RU^V OQ]PKV T=BD T:9V T=BD TTUP

n3/6S]jS`[jS^&=hF ^h<\!]2=OVe-D TN@T | eg]hQT` AnxÅRb ; ReS^jgOiVB=T

^ y ;

JG, R ;4I098 <)2B2BU@>#Mu<T;I>@?A:B8vNufuSt T|S\D .~S n<! 3 |!~SC

b­P^VR9Ij-=?>R6>@TD D TP!V HJIgTfh^V=?>` ]hLIVOi]!Pw@GF IgPl`BA`yV?C6D TfhQOiPK>^OL=TfPgTP->RUT``OiV TjS^`»hiTYRU^hQR6IghTfjghQOQRUOVT@gT8h1F OiP9R!T6=`T~@GF IgPgT~D8^ V?= ORT ;2 IS^PK@ ]!P ^ ]4T` ]OQPFP->^P-D ]OQPS`N@Th^RU^hQR6IghiT= ]!PFj_TIVcj-=]R>@gT6=u@T8h^D8^PgO<C= T`BIgOLRª^PV T ]_^`B>UT`[IK=hQ^/bÅ^RV ]7= O` ^V OQ]P LU

@gThQ^ D8^ V?= ORT0% | ^hRIghQT6=hQT`bÅ^!R VT6I-=`LU

@ThQ^ D8^ V?= ORT0%PA = LUZE>` ]2IQ@T= TlhiT`

n`[A`VBC6D T`hQOiPK>^OL=T`

LUxi = Pei ] ei 1 ≤ i ≤, n T`V­hQT iZ C6D TwR!TR VT6I-=O@ThQ^

]S^` TRU^Pg]PgOHJIgTN@TRn ; »^` ]hLIVOi]!P xi T`V[hQ^ i

ZCD T RU]hQ]PgPSTO@gTA−1 ;TRU]V­V ]V^h^jgj_^&=TUPVT`VM@gT

n3/3 + n3 = 4n3/3S]jS` ; km^O`]!Pmj4T6IVOD ]PVB=T6=MH7IST \2= RT 9h^`VB=?ISRVBI-=T jS^&= V ORIghQOLC6=TO@gT`` TR]!PK@g`D T6D/]-=T`E@T`[`[A`VBCDT`YhiOQP->^OL= T``[ISRRT` ` O bÅ` hQTR]gV=?>UThGP!F T`VH7IST@gT

n3 _]jS`Nx R b ; T6fT6=RORT(y ;r PgT^jgj-=]RegTV>HJIgOLR ^hQTUPV TR]7I-=^2D DTPV IV OQhQOQ`B>UT D8^OQ`4H7ISO4PSTjS^!` ` T[jS^!`fjS^2=hQT­R^hRIgh@T`SbÅ^!R V TI-=`LU T`yVh^D >Veg]T@T@OVT@gTv ^2IS` `[ZÀ\]7=?@g^PNH7ISO RU]PS` O`yVT9­jgO<R!]VT6=R]2D jghLCVT6D TUPVhiT`[A`VBCDTjS^2=^2D >UVB=?>

Ax− y = 0.

b­PjgOLR]V TYRTV V Tb ]O`q`BI-=qhQ^R]!hi]!PgPgTTUPV OLC6=TE@TbÅ^ U]P 9­V?=^PS` b ]7=BD T=»hQT`[A`VBC6D TYTUP IgP `[A`VBCDTE@O^&\!]PS^hx−A−1y = 0

; b­PDj4T6IgVY]7]S` T6=?RT6=HJIgTRUTV V T­V TReSPgOLHJIgT R]!PS` OQ`V T19cT4TR VBIST6=YTUP-jS^&=^hQh<ChiT19jS^2=VO<=E@Th^D8^VB=OQRUTO@TP!VOV?>hiT`YjgOLR]V^&\!T`P->RUT`` ^O<=T` 9h^cV?=^PS`[b ]2=?D ^V OQ]Po@TATUPhQ^ D8^ V?= ORT O@TUPVOV?> ;

vNufu3v T|S\D D z\o ~p| WOQTUPwHJIgTqV?=BC`GOLDj4]2= V^PVjS^&= `T`GPg]7Dw]-=T6IS` T`!IV OQhiO`^ V OQ]P_`TUPXPS^hLA`TfTVTP/vM>U]7D >UVB=OiTS@gO>6=TUPV OQTUhQhQTTUV4]gOQTUP H7ISTV= TD8^&=zH7I_^&]ghQTjS^&=`T`j-=]jK= OL>V?>`S>UV ]!PgPS^PVT` @S^PS`fhQ^ Ve->U]7= OQT@T` D8^VB=OQRUT` hQT@T>UV T6=?D OiP_^PVT`yVu@OROQhiT 9o@T> #SPgOL=TV 9KTfjghQ]OiV T=O@S^PS` hQT8RU^2@-= T @T8RT8R]7I-=` ; T @->V T=BD OQPS^PV T`VhQ^ R ^hQT6I-=u@!F IgPSTb ]7=BD T DwIghiV OQhiOQP->^OL= T `BI-=oIgPTPS` T6Dw]ShiT @T

nR!TR VT6I-=` HJIgOReS^P-\T @T ` O<\!PgT 9>ReS^2HJIgT j_T=BD/IV^V OQ]P

@gT`MRTR V TI-=`TUVNHJIgO R ^2IV : j4]2I-= hQT`nR!TR VT6I-=`M@GF ISPgT ]S^` Tc]7=Veg]Pg]7=BD >T ;!| F T`yV^&IS`` O hQT R]hLI-D T @TI

nZwjS^2=^hihL>UhL>UjgOQj*C@T8TP-\TPK@T=?>8jS^&=

nR!TR VT6I-=`u@g^PS`

Rn ; b­P `T ]4]2=PgT=^ORO9=^jgj_ThiT=hQT`jK= OQPSROQjS^hiT`jK= ]!j-= OL>V?>`M@TI @->V T=BD OQPS^PV TV:9 @g]PgPgT=­hQThiOQTUP´T``TPV OQTUh^XRTRchQ^ D >Veg]T@T @gT v ^2IS` ` R&F T`V[Z9Z @OL= T hQT@->V T=BD OQPS^PVT`VV>6\^h xÅ^2I `OL\PgT j-=?C`zy^&IKj-=]9@-IgOV@gT`YjgOLR]V` ; +#'17(9p %" %7(8JXj ')%+-,TE@T>UV T6=?D OiP_^PV@->Uj4TUPK@lhiOQP->^OL= TDTPV@TReS^7HJIgThQOL\PgTY`[>jS^&=?>6D TP!V ;X h `F TUPS`BIgOiV HJIgT

det(A+B) 6=det(A) + det(B)

TUVdet(aA) 6= a det(A)

; oPKT ¦TV 9lRe_^2HJIgTb ]O`EH7I F ]!P D/IghVOijShiOQTOIgPgT hiOL\PSTj_^&=a hiTu@->V T=BD OQPS^PVT`VVDwIghiV OQjghQO<>j_^&= a P @]PSR det(aA) = an det(A)

;,Tu@T>UV T=BD OQPS^PV[ReS^PK\TN@T`OL\PST ` O]!P-j4T6=?DwIVTu@T6I-fDhiOL\PST` ;det(I) = 1det(A) = 0 ⇔ A

T`yV` OQP-\2IghQOLC6=T ;det(AB) = det(A) det(B)

x)@g]PSRdet(A−1) = (det(A))−1 ;

det(AT ) = det(A); Y O

T = (tij)T`VV?= O^PK\2Igh^OL= T

det(T ) =n∏

i=1

tii.

Page 34: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

P&U

7(8JXj ')%+- <"%%8jk+T6'1$28 <QX #8 gTR^hRIgh @-Io@T>UV T=BD OQPS^PV@GF IgPgTND8^ V?= ORT

2× 2T`yVE]gOQTUP RU]PgP9I %

det

[

a bc d

]

= ad− bcx -;<:U y

RTuHJIgOGj4T6=?D TV@Tu@->V T=BD OQPgT6=T6fjghiOROiV TD TUPV[h1F OQPJR!T6=`TO@GF IgPgTND8^ V?= ORT2× 2

%

[

a bc d

]−1

=1

ad− bc

[

d −b−c a

]

.x T;L:(W y

^\7>UP->=^hiO`^ V OQ]P @T`Nb ]2=?DwIghQT`~x T;L:6U y Zzx T;L:(W y­^2I RU^`N@gT`ND8^ V?= ORT`n × n

RU]PK@TISOVc^2ITfnbÅ^2DTIS` T`b ]2=?DwIShiT`M@gT | =^&D T=OHJIgT Pg]7IS`PgT = Tj-=]9@-IgO<=]P_`jS^!`OQRUORU^&= ThihQT` ]!P!VOIgPgT8RU]2D jghQTfOV?> Tfj_]!PgTUPVOiThihQT RT/H7ISOhiT`E=TUPK@KOLDjK=^V ORU^&]ShiT`Yj_]7I-=@T`V@OLDTPS` Oi]!PS`YVB=?C`[j_TUV OiV T` ; X V OiVB=Tu@GF T6fT6D jghQT j4]2IK=[RU^hRIShiT=[hiT@T>UV T6=?D OiP_^PV@GF IgPgTMD8^ V?= ORT

20× 20jS^&=hQ^ub ]2=?DwIghQT@gT | =^&D T6=Oih*bÅ^&IgV9j4T6I-j-=?C` :XWUJd2d ^P_`4@TRU^hQR6Igh`BI-=NIgPgT~D8^ReSOiPgT~@T :(d2d k OQjS` xÅ` ]OiV108 OQPS`VB=?ISR VOi]!PS` jS^2=`TR]PQ@T(y ; X R!TRhQ^D >UV eg]T@T~@T`jgOLR]V` hiTR]V[P!F T`VHJIgTN@T

3 · 10−5 ` TR]!PK@T`oPj-=^ VOLHJIgT ]PRU^hQR6IghQT6=^chQTu@T>VT6=?D OiPS^PV^j-=?C`YjgOLR]V^&\!T %

det(A) = (−1)pn∏

i=1

uii

] hQT`uii

x1 ≤ i ≤ n

yc`]!PVhiT`jgOLR]V`TUVphQTPg]7Dw]-=T @Tj4T6=?DwIV^ VOi]!PS`T4TR VBIK>UT`l^&IIRU]2I-=`u@Th^

bÅ^RV ]7= O` ^V OQ]P ;J, 034=Mu>OMY0 87$7')$28 g1 ' p(p89j %01+ *8o %889pp(KX(p@ :2; | ^hRIShiT=qhiT`bÅ^RV T6IK=` r @T[h^D8^VB=OQRUT

A`[IgOLR ^PV T

A =

2 −1 0 0−1 2 −1 0

0 −1 2 −10 0 −1 2

.

oPt@T>(@TIgOL= T hQT`bÅ^!R V TI-=`LU

@!F IgPSTND ^VB=OQRUT`[A9D >V?= OHJIgTA n× n

aii = 2

ai,i+1 = −1

aij = 0, ∀j > i + 1.

T; b­P RU]PS` OL@-C6=Tn + 1

=T``]7=V`E@TwD0,DTchi]!P-\2IgTI-=LTUVM@TcRU]PS`V^PV Tu@Tu=^O@TI-=

k ^!` ` T6D/]ghL>`]4]2IV:9 ]4]2IVTV#-fT>`^&ITft@TITf Tf VB=?>6D OiVB>`@ThQ^Re_^ PgT ;!| eS^2HJIgTcPg] T6IK@iT`yV ^``]ROL> 9 IgPgTub ]7=RUT

fi@OL= OL\2>T`Thi]!PKhF ^fTN@T`=T``]7=V`TV[]PPgTRU]PS` O@TC6=TMHJIgThQT`@T>Ujgh^RUT6D TUPV`Yhi]!P-\OiVBIK@gOiPS^2ITf @TI `BA`yV?C6D T ;b­PPg]VTuihQTu@T>UjShQ^!RT6D TP!VV@TIPg] TIK@

i;. Th^/=TUh^ VOi]!P D >RU^PSOLHJIgT

f =k

Lδu,

]PKVO<=T hiTN]gOQh^P@T`T4]7=V`^&IKPg] T6IK@i

k

L(ui − ui−1)−

k

L(ui+1 − ui) + fi = 0.

Page 35: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

PJWb­P^ ^&ITfKTf VB=?>6D OiVB>` u0 = un+1 = 0

;k TUVV?= TRUT[`[A`VBCDTV@!F >(H7I_^ V OQ]P_``]7IS`qh^Mb ]2=?D TAx = b ] A

T`V IgPSTED ^VB=OQRUTn×n

TVbIgP~RTR V TI-=

@gTRn HJIgThF ]!P-T6f jShiOROiV T=^ ;oPHVTUPS^P!VDRU]2D jVT@-IHjK= ]"#ShE@T

A @g]PgPgT= hiT Pg]7Dw]-=T@T S]j_`lPK>RT` `^OL=T`09´hQ^ =?>` ]hLIV OQ]P @TI`BA`yV?C6D TN@T`V= T` ` ]2= V` ;

8J$2')$78 gFg 35+-6')$78t #8'10 8JX@ ` = O^P-\7IghQT6=jS^2=fh^uD >Veg]T@T@gTMv ^&I_` ` h^ND8^ V?= ORTM@T­nOihL]4T6= V3× 3

`[IgOLR ^PV T

H =

1 12

13

12

13

14

13

14

15

N>6=O #_T6=H7IST­hQTM@->V T=BD OQPS^PVE@THT`V>6\!^h4^2Ij-=]T@TIgOiV@T`jgO<R!]V``BISRURUT``O<bÅ` ;-| ^hQR6IghQT6=h1F OiP9R!T6=`T@gT

H;

8J$2')$78 gFL ! %"#') j +T6')$78 7*8J$289"# #+K#pRn Y ]!OV

aTV

b @T6I-fMR!TR VT6I-=`G@T Rn ; b­Pu@T> #SPgOiVh^ D8^ VB=ORTA (n× n)

jS^2=aii = 1 + a2

i + b2i

aij = aiaj + bibj ,`O

i 6= j.

^Jyo fj-= OLD T6=h^ D ^VB=OQRUTATUPob ]PSRV OQ]P@T

InTV@gT

a b aT bT ;]Qy8b­P @T>` OL= TKT ¦TRVBIgT=hQTj-=]9@-IgOVAx x ∈ Rn ; R=O<=ToIgPH^h<\!]2=OVe-D T =?>^hiO` ^PVRTj-=]T@TIgOiVTP

O(kn)S]j_` ] k

T`VVIgPobÅ^RV T6IK=9 @->V T=BD OQPgT6= ;

Page 36: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

P2^

Page 37: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

)#1) (

,.+ g:;4I?GCBR M ;=>.?A:T`c^h<\!]2=OVe-D T`N@T>R=OV`u@g^PS`RT-R]2IK=` TUVcTUP jS^2=VOQR6IghQOiT=cRTITf @TIIRe_^jgOiVB=T b ]PVcOQPV T6=?RTPgOL=NIgPRUT6= V^OiPmPS]2Dw]K= T @GF ]!jQ>=^V OQ]PS`>UhL>6D TP!V^OL= T`@T`yVOiPK>UT` 9 ,VB=TlVB=^OiVB>UT`j_^&=MIgP ]7=?@OQPS^V T6IK= ;G| eS^2HJIgT]2=BZ@gOiPS^V TI-=^ IgPgT D8^PgOLC6=TcjK= ]!j-= T @T = Tj-=B>`TPV T6=­hiT`PS]2Dw]K= T`=?>UThQ`­TV h1F TUP_`TDw]ghQT @gT`=B>TUh`HJI!F Oihqj4T6IgV

=TUjK=B>`TP!VT6=YT`V#SPSO h^8RhL>j4]2I-=­R]2D j-=TUPQ@T= T h^8`yV?=BI_R VBIK= TN@TcRUT`]7IS`[ZtTPS`TDw]ghQTN@Th^~@T= ]!OVTu@T`V=B>TUh`>UV^P!Vh^ j-=?>RO` Oi]!P D8^RegOQPgT ; F ^2= OiV e-D >UV OH7ISTTUPj-=?>RUOQ` Oi]!P #SPgOQTTVh^ =?>^hQOiVB>u@T`V@]PSP->UT`OiPSTfg^RV T`[Pg]2IS`[]2]ghQO<\!TUPV98R]!PS` OL@T>= T=hQT`EHJIgT`yVOi]!PS``BIgO<R ^P!VT`=TUhQO<>T`9h^/=?>` ]hLIVOi]!Po@GF IgP `BA `VBCD ThQOQP->^O<=TAx = b

%`OATUV

b` ]PVqj_T=V?I-=B]*>` jS^2= IgPgTOF j_TUV OiV TVHJIS^PVOV?>2F R]2D D TUPVqhQT`q` ]hLIV OQ]PS` T6f^!R VT` x

TVRU^hQR6IghL>UT`xc` ]PVTUhQhiT`^ ¦TRVB>UT` ,HJIgT`OL\PSO #ST P9I-D >6=OH7IST6D TUPVhiTObÅ^OVHJIgT

AT`yV[j-=T`?HJIgT `OQP-\2IShiOLC6=T `O

b /∈ Im(A) R]7D DTPVE@->V T=BD OQPgT6= xj4]2IK=HJIgT

Ax`]!OV`BI `^&D D TUPVNF j-=]RegT2F-@gT

bb­Pl`6F ^ V V^&=z@T=^YORO 9V@]!PgPgT=%HJIgThLHJIgT` >h<>DTPV`%@T =?>Uj4]PS` Tf^2ITfu@T6ITfjK= TDOLC6=T`#HJIgT`V OQ]PS` h^V?= ]!OQ` OLC6D T>UV^P!VjghLIS`j_^&= V ORIghQO<C= TD TUPVYVB=^OiVB>UTN@g^PS`YhQTReS^jgOiVB=T `[IgOLR ^PV ;

,KJ ?)4 JL098DJL<T;47>OMu>.09U@U@09810Y; Mu?A:BCB>r;=>@?A:C NRB:50 JL<T;47>OMY0Nu3vfuSt |xDxm<S 8 <y 6 < <X6!~ 0S 0

r PgTcPg]7=BD TuR!TRV ]2=OQTUhQhiTT`yVISPgTub ]!PSR VOi]!P Pg]V?>UT ‖.‖ @T> #SPgOQTl`BI-=ISP T` jS^RUTuRTR V ]7= OQTUhTV`^ VOQ`[bÅ^O`^PV^2ITfV?= ]!OQ`^&f OQ]2D T``BIgOLR ^PV` %:2; ‖ x ‖≥ 0

j4]2I-=YV]2IVxTUV ‖ x ‖= 0⇔ x = 0

T; ‖ ax ‖= |a| ‖ x ‖ j4]2IK=YV ]2IgV xTVV ]7IV`RU^h^OL=T

aP-; ‖ x + y ‖≤‖ x ‖‖ y ‖ j_]7I-=YV ]7IS` x, y

89j +0)8F1 D?*Rn &BF!e;CO>F!I"6Oa?U?9

xT y c

‖ x ‖2=(

xT x)

1

2 =

n∑

i=1

x2i

"NXDX>?p&( m7 O#"=UAX>@K&(D?ap9.:UFX c km;CIaO>8?& &('C"n'&e;C>OFG ‖ x ‖∞= max

1≤i≤n|xi|

O#"U`X>@K&(e! & yqU> X>@K&(d80oa rX^O\a rX?_ c > X>@K&(l∞

s

P9_

Page 38: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

P2

‖ x ‖1=n∑

i=1

|xi|O#"U`X>@K&(

l1

UOd".O>@kfX>@K&(6;COG,aOG,8X"%f>@C"8da6;o"3.a,9UKD8OfX>@K&(lp ‖ x ‖p=

(

n∑

i=1

|xi|p)

1

p

SF OQP->6\^hQOV?> V?= O^P-\7Igh^OL= T RU]2=?= T`j4]PQ@g^PV T T`VoIgP =?>`BIghV^ VKO<D j4]2= V^PV@T´h1F ^PS^h<A` Ttb ]!PSR VOi]!PgPgTUhQhQT @$9Fk OQP ] [` OTUV` ^ @T>6D ]!PS`yV?=^V OQ]P @T>RU]2IghQTo@gTKhF OiP->\!^hiOiVB>ob ]PQ@g^&D TUPV^hQTK`BIgOLRª^PV T@OiV TOiPK>6\!^hiOiVB>@T*0) %87 %

(∀p, q > 1 /1

p+

1

q= 1)

n∑

i=1

|xiyi| ≤‖ x ‖p‖ y ‖q .

b­P ]2]_`T=BR!T6=^ >\!^hiTDTPVwH7IST @g^PS`Rn V ]7IV T`hQT`cPg]2=?D T`c` ]PVu>(H7ISO<R ^hQTUPVT`w@S^PS`chiTD` TUP_`] j4]2I-=@TITf-Pg]7=BD T` ‖ . ‖a

TV ‖ . ‖b OQhGTfO`yVTu@T6ITfKR]P_`yV^PV T`Yj_]`OiV OLRT` αTV

β`^ V O`[bÅ^O` ^P!V

α ‖ x ‖a≤‖ x ‖b≤ β ‖ x ‖a, ∀x ∈ Rn.

fu3vNu3v 6 0 |!~p\~ <p <b­P/@->/#SPgOiV D8^OQP!VTUPS^PV%@T`»PS]2=?DT``[I-= hF T`jS^!RT@T`%D8^ V?= ORT`»R^&=?=B>T`

(n×n); b­PlTfOL\T4@gTjSh<IS`HJIgTRT`Pg]7=BD T` R2>= O$#STUPVSIgPSTR]!PK@OiV OQ]P`BIgjgjghL>6D TUPV^OL= T @OiV TRU]PK@OiV OQ]P @gT[Pg]2=?D T p*"%pzj "%0('8+0)')$7+T'*7Q8 %

‖ AB ‖≤‖ A ‖ · ‖ B ‖d]2I-= >6R ^hLIgT6=qhQT`T ¦TV`4@GF IgPgTYVB=^P_` b ]7=BD8^ VOi]!PlhQOQP->^O<=T`[I-=fhQ^Pg]7=BD TV@GF IgP RTR V TI-= ]P8`F ^2=B=^P-\!T6=^j4]2I-=IVOihQOQ` T6=@T`YPg]7=BD T`E@TOD ^VB=OQRUT` p(" 4K %Q%#989p ^&ITf-Pg]2=?D T`RTR V]2=OiThihQT` @g^PS`YhQT` TUPS``BIgOLRª^PV %

‖ A ‖= supx6=0

‖ Ax ‖‖ x ‖ = sup

‖x‖=1

‖ Ax ‖

] ‖ A ‖ @T>/#SPSOVOIgPgTlPg]2=?D Tl`BI-]_]7=?@g]PgP->T 9h^PS]2=?DTwRTRV ]7= OQTUhQhiTwIV OQhiO`B>UT @g^PS`­hQTlRU^hQR6Igh@T`­Pg]2=?D T`RTR V]2=OiThihQT` ‖ x ‖ TUV ‖ Ax ‖ ; b­Pt=TV?= ]7I-RTlhQ^ @T>/#_PgOVOi]!P I_`[IgThihQT/@!F IgPSTPS]2=?DT @GF ^jgjghQORU^ VOi]!P hQOiPK>^OL=T ;b­P ^ ^hQ]2=`h^ = ThQ^V OQ]P ‖ Ax ‖≤‖ A ‖‖ x ‖ j_]7I-=YV ]7IV x;b]S` T6=?R ^ V OQ]P_`%

:7; hiT `[ISj´RUO Z @T``BIS`@g]OiV1,UVB=Tlj-=O`M@S^PS`Cn D8^O`RU]tPSRO@T^XRTRchiT `BIgj @S^PS` Rn j4]2I-= hQT`Pg]7=BD T`

l1, l2TUV

l∞;

-; jS^&=h^ R]2D jS^!ROiVB>N@Th^ `jgeKC6=TMISPgOV?> hQT`BIgj-T`yV^VVTUOQP!Vj4]2I-=IgP xPg]PKP9Igh

+%')J9p~ %89p %QXj 8Jp p(" Q #*%#989p:7; ‖ A ‖= inf ρ/ ‖ Ax ‖≤ ρ ‖ x ‖

-; ‖ I ‖= 1

8Jj +%018g kDX>?p&(6& F".K.aKUn#!^>@8>?FXG# k X>?p&(f@aK3'FCO#" ‖ A ‖2= [λmax]1

2 c> #λmax

O#"Ud;C3:fl:?\m_O?@n;C>#;COd8AT A c apgCa rX#;

>@ _@Gp9EX(AHX>@K&(H& F".K.aKUH#!^>@8>?FXG#OH X>?p&(Ol1

@"l∞

'>?X"e8>@FCG#;o?‖ A ‖1= max

j

i

|aij |?" ‖ A ‖∞= max

i

j

|aij |

Page 39: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

j7 P7c kAC>@K& d?ap9.:UFXeG?"I?C"f ap9U a?3a' c >@ IX;COG'g3$D>8F_?'C"=k`X>?p&(H8DUB".8aOm>8

X>@K&(e8 FO>^'FI

‖ A ‖F =(

Tr(AT A))

1

2 =

i

j

a2ij

1

2

.

6@"p"bDX>?p&( 7 "6#!^>@8>?FXG# FaX X>?p&(H_@@a"%>@KURn

b­Pj4T6IVVR7>6=O #ST=EH7IST j4]2I-=EISPgTND8^ VB=ORT ]2= V eg]7\]!PS^hQT Hxi.e.

VTUhQhiTNHJIgTHT H = HHT = I

y ]!PK^‖ H ‖2 = 1

‖ H ‖F =√

n.

i 89jk+TG"#8) 9U 'C".O ‖ A ‖2@"=m_OU?K;CO>O;\O

A'';COG,apkGDF(a rX#;CI".O

TN=B>`[IghiV^V[`[IgOLR ^PVED ]PVB=T h1F IV OQhQOV?>j-=^ V OHJIgTN@T`YPS]2=?DT`D8^ VB=OROQTUhQhiT`% #9Q@9jk8) !>@I" ‖ . ‖ CX>@K&( & ".K3ap'9U!^>@8>@FCG#f@"

EX& "3p3aD"%'9U ‖ E ‖< 1

U>?KfU`& ".K3a

A = I + E"C_@Kpb^?U

g g RU]PS` O@T>6=]PS`­hQT`[A`VBC6D T eg]7D]7\2CPgTAx = 0

;G hf`F >R=OiVx + Ex = 0 TVc^OiPS` O

‖ x ‖=‖ Ex ‖≤‖ E ‖‖ x ‖ RU^&=­hQ^Pg]2=?D TT`V`BI-]_]7=?@g]PgP->T ;*| ]7D DT ‖ E ‖< 1 ]!Pm^8P->RUT`` ^O<=T6D TP!Vx = 0

TVAT`V@]!PSRPg]P`OQP-\7IghiOLC6=T ;

Nu3vfu Tn ~%F~Sn . |0!~pC

V?IK@OQ]PS`»hQT`!R ^2= O^ VOi]!PS`G@Th^[` ]hLIV OQ]PN@GF IgP`BA `VBCD T @GF >HJIS^ VOi]!PS`GhQOiPK>^OL=T`!HJIS^PK@ hiT`TR]PQ@D TDw]-=T`BI-]gOiVVIgPgTj4T6= VBIK=B]S^V OQ]P %Ax = b⇒ A(x + δx) = b + δb

b­P´j4T6IVu@]!PSR >R=OL= Tδx = A−1δb

@GF ]´hiT`O= ThQ^V OQ]PS`>R6= OiV T`^XRTRw@T` Pg]2=?D T` `[IK]_]7=?@]!PgP->T` ^jgjK= ]2ZjK= OL>UT` %‖ δx ‖≤‖ A−1 ‖‖ δb ‖‖ A ‖‖ x ‖≥‖ b ‖TUVc]PFj4T6IV/>R=OL= T hF T`yVO<D8^ VOi]!P @T8h1F T6=?=T6I-=N=TUh^ VO<R!TTP Pg]2=?D T8`[I-=

xTUP b ]P_R V OQ]Pk@gT8hQ^j4T6= VBI-=?]S^V OQ]P

=TUh^ VO<R!T`BI-=b%

‖ δx ‖‖ x ‖ ≤‖ A ‖‖ A−1 ‖ ‖ δb ‖

‖ b ‖»^ H7I_^PV OiVB> ‖ A ‖‖ A−1 ‖ T`V^jSj_Th<>Th^ $2*# %'(')* @Th^/D8^VB=OQRUTT`yVPg]VB>T σ(A)

;

+#'17(n) σ(A) ≥ 1

63:DU a>@o3"3U>?O#" l:?\ c ;\3U & F".K.aOH"D3"%m& aO>@\:".>@FCG# c ]_~aZ ~ ;CIN%!"b$&

Ax = b"!"I!^?

σ(A) = σ(A−1)σ(aA) = σ(A)

;>m"b>F"6Oa?U?9a 6= 0

8

H"fXD& "3p3ad>?"3rX>,lF>@*U

σ(H) = 1

Page 40: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

U7d

fu3vNu Tn D~%F~ 0nxmn | 6 0 \S~ ~S< ^RU]PK@gOVOi]!P8T`VqVB=?C`hQOL>UT[^&I-f R ^hQT6IK=`j-=]jK= T`qTf VB= ,6D T`S@T

A TVf` OSh^ Pg]7=BD TVIVOihQO`[>TT`VfhQ^Pg]2=?D TT6I_RhQOL@OQTUPSPgT ]!PR2>6=O$#STMHJIgT&%σ(A) =

(

λ (AT A)

λ (AT A)

)

1

2

@T` ]2= V TNHJIgT`OAT`V`BA9D >VB=OH7IST

σ(A) =λ (A)

λ (A)TR^hRIgh#@ThQ^ R ^hiTI-=Tfg^RV Tu@Tσ(A)

T`yVRU]VT6ITf-TVOQhT`V[j4]!``OL]ghQTu@GF T` `^XAT= i eXV]_mkXV] @TN=?>@TIgOL=ThiT=OQ`?HJIgTN@GF T=B=T6IK= ;hMG"#'10)'%+ *8

| F T`VwIgPgT D >UV eg]T@T-j-=^2\2D8^ VOLHJIgT~HJIgOP!F T`VcjS^`u>6R OL@gTUPV T 9tDTUVV?= TTP ] T6I-R9=T ;%Y ]P j-=OiP_ROQj_TT`yV@GF T`` ^XA!T6=u@GF >HJIgOQhQO<]-=T6=chiTj4]O@g`/@T8V ]7IS`hQT`lR] T RUOiTP!V`u@Th^D8^ V?= ORT ;! hSbÅ^2IVcV?= ]7I-RT=u@T`wD8^ VB=ORT`@O^&\!]PS^hiT`

D(1) = (d(1)ii , 1 ≤ i ≤ n)

TUVD(2) VTUhQhiT` HJIgT σ(D(1)AD(2)−1

) << σ(A) h^ D8^ VB=ORTD(1)AD(2)−1 `F >R=O<R ^PV %

D(1)AD(2)−1=

(

aijd(1)ii

d(2)jj

)

b­P=B>`]7IV^hQ]2=`

D(1)AD(2)−1y = D(1)b

D(2)x = y| TP!F T`V[jS^`V]2I y]7I-=` bÅ^!ROQhiT TV[j_^`V ]7I y]2IK=`YT8R^RUTi +k#89jk89/')J+T'

b­P´j_TIVc`[ISjgj_]`T=uHJIgT hQ^Kj4T6= VBI-=?]S^ VOi]!P @-IgT^&ITf T6=?= TI-=`M@GF ^&=?= ]!PK@O`PgT8j_]7=VT~H7IST `BI-=hQT8`TR]PQ@D T6D/]-= T ; b­P^lj_^&=TfTDjShiTO=?>^hQO`[> TUPoD8^RegOQPgTOIgPgTObÅ^!R V]2=OQ`^ VOi]!P

LcUc = A + E

TV[]!P ^/=?>` ]hLIDT6fg^R VT6D TUPVhQT`[A`VBC6D T(A + E)xc = bb­PR2>= O$#STM\2>UPK>6=^hQT6D TUPVV@g^P_`RTRU^!`EH7ISThQTN=?>` OL@TIRU^hQR6IghL>

rc = b−AxcP F T`yV[jS^!`YP9Igh ;d]!` ]PS`^hQ]2=`

x(0) = xcTV

r(0) = rc;| T`VRª^hiTI-=`OiPSOVOQ^hiO`TPVIgP j-=]RT``BIS`=B>RI-=` O b ; b­PmR^hRIghQTj4]2I-=V ]7IV

k ≥ 0

(A + E)e(k+1) = r(k)

x(k+1) =(

x(k) + e(k+1))

c

r(k+1) =(

b−Ax(k+1))

coPm`BIgjgj4]!`^PVVHJIgTh1F T6=?=T6I-=V@GF ^&=?=]PK@O¦j4]2= V TT` ` TUPV OQTUhQhQT6D TUPV`BI-=hQ^ =?>` ]hLIVOi]!Po@T``[A`VBCDT`hQOiP->^OL= T` ]Pj4]!` Tx(k+1) − x(k) = (A + E)(−1)r(k)

e(k+1) = (A + E)(−1)A(x − x(k))

@GF ]hF ]!PDVO<=Tx− x(k+1) = (I − (A + E)(−1)A)(x − x(k))

Page 41: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

UQ:d]`]!PS`

G = I − (A + E)(−1)A = I − (I + A−1E)−1 ]P^ @g]PSR ]7]V TPJI j_]7I-=YV ]7IV k ≥ 1

x− x(k+1) = G(x− x(k)) = Gk+1(x − x(0))

Y Igjgj4]!` ]PS`^hi]7=`GHJIgTqj4]2IK= IgPgTqPS]2=?DT`BI-]4]2=z@]PSP->UTq]!P ^OV ‖A−1E‖ = α < 1 ^hQ]2=` G =

∞∑

k=1

(−A−1E)k

T`yVVTUh!HJIgT ‖G‖ < α1−α

TUVx− x(k) ≤

(

α

1− α

)k

(x− x(0))

F T6fjQ>= OQTUPSRUTj-=]2I-R!TVH7I F IgPST[]2I~@TITfOiVB>=^V OQ]PS` @TRTj-= ]RUT``[IS`j4T6IKRTUPV,UVB=TOiPV?>6=T`` ^P!VT` D8^O`jS^!`jSh<IS` ; Z.X[q.]_qZ iezegow]_qrivbf]_mv`b­PR]!PS` OL@TC= T hQT`[A`VBC6D T`BIgO<R ^P!V %

(S) :

10x + 7y + 8z + 7w = 327x + 5y + 6z + 5w = 238x + 6y + 10z + 9w = 337x + 5y + 9z + 10w = 31

@g]PVh^`]!h<IVOi]!PDT`yV^jgj_^&=TUPV T %x = y = z = w = 1

; ZTR^hRIghQT6=hQ^`]!h<IgV OQ]P-^XR!TRIgP-` TRU]PK@ DTDw]-=Tj4T6= VBIK=B]*>N] x P T;L: T; c/P7P-;L:MP7d-; c y T ;2 IST =T6D8^&=zHJIgT 6ZR!]2IS` | ^hRIghQT6=lhQ^ R]!PK@OiV OQ]P @T-hQ^nD8^ VB=ORT @-I>`[A`VBCDT x ]!PkIV OQhQOQ` T6=^mh^mPg]7=BD T @gT = ]7]_TPgOLIS`YTV[]!P IS`V O$#ST=^8RUT ReS]O<f-y ;

, L >r;=098=80 CB0 MY?A: A034 A09:BMY0 CB098D8RB>r;=098r PgT @T`O@ORIghiVB>`u@T h^-RU]PS`VB=?ISR VOi]!P @GF ^hL\]7= OiV eKDT`T`VhQT8R]PV?=!hiT @T hQ^KR]!P9RT6=?\TPSRT R2F T`yVBZ 9&Z@gO<=T j_]7I-R]!O<=­OL@TPV O$#ST6= `O»hQ^-`[ISOVT/@T`­` ]hLIV OQ]P_`RU^hQR6IghL>UT`9ReS^7H7ISTOV?>6=^ VOi]!PmR]P9R!T6=?\T ]2Imj_^` TV `O]7IgO ^XRTRMHJIgTUhQhQTNROiV T` ` T`6F ^jSj-= ]RegT6Z VBZtThihQT@gThQ^ ` ]hLIV OQ]P@TIKj-= ]7]ghLC6D T ; b­P@OL=^ HJI!F IgPgT`BIgOVT xk

@gTRn R]P9R!T6=?\TERT=` x ∈ Rn ` O4h^c`BIgOiV TO@TM=B>TUh` rk =‖ xk − x ‖ RU]P9RT=B\!T @S^PS` R RT=` d-; SF >HJIgOLR ^hQTUPSRUT@gT`YPg]7=BD TOD ]PVB=TNH7ISTh^ Pg]V OQ]P@TR]!PJR!T6=?\TPSRT­T`VYOQPK@T>j_TPK@g^P!VTu@TI Reg]O<f @T h^ Pg]2=?D TM@S^PS`

Rn ; 4%%'(')* F1 8t %8t$7Q 7*87 *89#$78 6>?!p.8G>?!NCI"bm8fOG#9 rk

a>?C_@'plD_?''X_O?@

r∗ ;F;='9UDZ5C5:4A8DaO>@C_@SlF'\aOf8fkdI"b rk

;CIl'\H'C".'p > 0

"b'0

limk→∞

sup|rk+1 − r∗||rk − r∗|p = β < +∞

»^>` OV?IS^ VOi]!P@T hQ]OQPh^IjghLIS`KR]7I-=^PV T T`Vp = 1 ^jSj_Th<>T R]!P9RT6=?\TPSRT hQOQP->^O<=T ;E| F T`yVKhiT´RU^`V A jgOHJIgT6D TP!V»j_]7I-=!IgPgT`[ISOVTS\2>]2D >V?= OHJIgT

rk = ak, 0 < a < 1; ^YhQO<D OiV T

βT`yV»hiTS=^XA]!PM@gTfR]!PJR!T6=?\TPSRT TUV@g^PS`hQTRU^`V\2>U]7D >VB=OH7IST ]PKV?= ]7I-RT IS`V TDTPV a

; . ]!PSR jgh<I_`[hQTN=^XA!]P @gTcR]!PJR!T6=?\TPSRTT`V[j-=] ReST@gT d jghLIS`h^cR]!P9RT6=?\TPSRTT`yV=^jgO@T/xÅ]PDj_^&=hiTM@T­RU]P9RT=B\!TUPSRUT[`[Igj4T6=hQOiP->^OL= TMHJIS^PK@ β = 0 RTOHJIgO_T`yVhQTlR^`jS^&=TfT6D jghQTcj4]2IK=h^D`BIgOVTrk = (1/k)k y ; . F ^&IgVB=TjS^&= V jghLIS`­hiTcV^&I-fT`yVj-=] ReST/@T : jSh<IS`h^RU]P9RT=B\!TUPSRUTT`yV[hQTUPV T~xÅjS^&=T6f TD jghiT

rk = (1/k)y ;

2 I_^PK@p = 2 ]PKjS^&=hiTN@TRU]P9RT=B\!TUPSRUTHJIS^2@-=^V OH7IST xÅjS^&=T6fT6D jghQT rk = a2k y ;

Page 42: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

U

, S;=< >@U@>r;=6 CB0 U@<&JL6Y; B?GC50$C50 <ARB88b­P ^ R9Imj-=?>R>(@T6D D TUPV h1F OQP$KIgTPSRT @TInD8^2I-R ^O`­RU]PK@OiV OQ]PSPgT6D TUPV`[IK=­h^D`V^2]gOihQOiVB> @Tlh^D` ]hLIVOi]!P

@GF IgP`[A`VBC6D TO@GF >HJIS^ VOi]!PS`hiOQP->^O<=T` ;J hQbÅ^2IVY]2]S` T6=?RT=4HJI!F IgP D8^&IKRª^OQ`YRU]PK@OiV OQ]PSPgT6D TUPVYPgT `OL\PgO$#ST­jS^`P->RT``^OL= TDTPV HJIgTKhQ^ D8^ VB=ORTKT`V HJIS^!`OY`OQP-\7IghiOLC6=T ; | F T`VhQT@T>VT6=?DOQPS^P!V~HJIgOD T`BI-= TRTUVVToHJIS^!`O` OiP-\7Igh^&=OV?>nHJIS^PK@ OQh `6F ^jgjK= ]RegTt@T d-; d^2=DT6f TD jghiT hQ^ D8^VB=OQRUT(n × n) aI ^XRTR a = 10−1 R]!OV` ]P @T>VT6=?D OiPS^PVV TUPQ@T= T~R!T6=` d HJIS^PK@

nV TPK@ RT=`hF OiP #_PgO ^hQ]2=`OH7ISTD` ^RU]PK@OiV OQ]P =T`V T~>\!^hQT&9 :2;^ ` OiVBIS^V OQ]P OiP9R!T6=`TKT`VTUPSRU]2=Tjgh<I_` _^&\7=^PV T^XRTRoIgPSTD ^VB=OQRUTPg]P `BAJD >UVB=OLHJIgT@g^PS`8hF T6f TD jghiTRO<Z @T``]7IS` %A =

[

1 1000 1

]

Y Ob =

[

1001

]

, x =

[

01

]

D8^O`[` Ob =

[

1000

]

, x =

[

1000

]

T@T>VT6=?DOQPS^P!V@TAT`yV]gOQTUP`-= >\!^h9 : j_]7I-=V^PVh^R]!PK@OiV OQ]P @TRUTV V TD8^VB=OQRUT[T`VVB=?C` D8^&I-R ^O` Tx

> 104 y ;T4D8^&I-R ^O` R]PQ@OVOi]!PgPgTDTPV@GF IgPgTD8^ V?= ORTT`V ` ]2I-R!TUPVV?=BC`#@OROQhiT 9V>R ^hLIgT6= TVq`[IK=V]2IV h^HJIgT` ZV OQ]P@TI=T6D C@gT 9/AK^jgj4]2= V T== T`yVTu@T>UhQOQR^ VT ;-` ]2IVTb ]O` RT=V^OQPgT`Y` OV?IS^ VOi]!PS`j4T6I-R!TUPV,VB=TObÅ^RUOihQT6D TP!VR]!PV ]2IK= P->T`6%R&F T`V hiT8RU^!` jS^&= T6f TD jghiT]´hQT` e->UVB>6=]2\7>UP->OV?>`­P9I-D >6=OLHJIgT`M@-I´`BA`yV?C6D T8^jgjS^2=^OQ`` TUPV`BIgO<R ^P!VhiT`hQO<\!PgT`c]2IIhQT`lR]!hi]!PgPgT` ; b­P j4T6IgV^hQ]2=`T4TR V?IgT6=/IgP>ReS^P-\TDTPV/@GF >RegThihQTnx p($2+K0)') yTV^&D >hiOQ]2=T6=h^mR]!PK@OiV OQ]P @Th^D8^ V?= ORT ; b­PI` TR]PVTUPV T=^KORO4@!F OQhihLIS`VB=T6=cRUT~bÅ^OiVl`BI-=uIgP>TfT6D jghQT VB=?C`` O<D jghQT&%

(S) :

10x1 + 100000x2 = 100000x1 + x2 = 2

r PST =?>` ]hLIV OQ]P @OL= TR V T @TII`[A`VBCDT8TPI^&=OVe-D >VOLHJIgT8V?= ]!PKHJI->UT 9KV?= ]!OQ`cRegO=T`c` OL\PgO$#_RU^V O<bÅ`Ob ]2IK= PgOL=^IgPgT`]!h<IgV OQ]Px x d-; d2d:7; d7d y ;*Y O»]!PDwIghiV OQjghQOiTchQ^8jK= TDOLC6=Tu>(H7I_^ V OQ]PjS^2=IgPbÅ^!R VT6I-=M@GF >RegTUhQhQTw@T

10−5]P]7]V OQTUPVh1F ^jSj-= ]XfOLD8^ V OQ]Px′ x :7; d7d :7; d7d y ]_T^&ISRU]2IgjKjghLIS`[`^ VOQ`[bÅ^O` ^PV T ; hQbÅ^&IVj_]7I-=fVT6=?D OiPgT= D TV VB=ThF ^!RURUTUPV`BI-=fhQTVbÅ^OVHJIgTD0,6D TIgPgTMD ^VB=OQRUTV]gOQTUPDRU]PK@gOVOi]!PgP->UT­T`V@T@T>UV T6=?D OiP_^PVfV?=BC``BIgj*>6=OiTI-=9 d j_TIVYj-= ]R!]TH7IST6= @T`4@gORIShV?>`PJIKD >= OHJIgT``O4hF ^h<\!]2=OVe-D T@GF >UhQO<D OQPS^ZV OQ]PT`yVVD8^hRU]PVB= hL> ;K| F T`V[hiTR^``BI-=V@T`[jgO<R!]V`fV?= ]!j-j4TVOV`` ]PV[^RRTUjgVB>`^2IKRU]2I-=`@T`RU^hRIShQ` ;

dS=TUPg]!PS`IgPTfTDjShiT` O<D jghQT&%

(S) :

0.0000x1 + x2 = 1x1 + x2 = 2

Y O ]PD ^OiPVOiTP!V d-; d2d7d-: RU]2D D T j-=T6D OQT6=KjgO<R!]VKTV `OhiT`-R^hRIgh`K`]!P!VKT ¦TRVBI->`K^XR!TRV?= ]!OQ`-RegO*=T`` O<\!PgO #4RU^ VO bÅ` ]Pm]7]V OQTUPK@-=^ x2 : TUV­h^D`BI-]S`V OiVBIVOi]!P ^&=?= OLC6=Tu@]!PgPgT6=^ dK; d7d2d-: x1 + 1 = 1 `]!OV x1 = 0 TV[hF ^&D jghQO #_R^ VOi]!Po@ThF T6=?= TI-=E@GF ^2=B=]PQ@O¦T`VRU^ V^`VB=]jgeSOLHJIgT ;

T = TD C@T 9 RTUVV To@O8R6IghiVB>KT`VR]!PgP9I%R&F T`Vlh^m`yV?=^VB>\OQT @TI +'87*K +%+-((')890 HJIgO=T6R OiTPVcOQRUO9j4T6=?DwIVT6=[hQT`E@TITfDhQO<\!PgT`YTUVR]7D D TPSRT=h1F >UhQO<D OQPS^ VOi]!PjS^&=RThihQTuHJIgOGj-=?>` TUPV T hQTjghLIS`E\2=^PQ@jgOLR]V ;

Page 43: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

UJP , 034=Mu>#Mu098 8J$2')$78n) | ]PS` OL@->6=]PS`hQT`[A`VBC6D T hQOiP->^OL= T

Ax = b@g^P_`

R4 ^XRTR

A =

1 2 0 −11 3 1 2−1 −2 2 −1−2 −1 3 11

, b =

111α

:2;| ^hRIghQT6=hQTN=^P-\ @Th^/D8^VB=OQRUTA;

T; d]2I-=HJIgTUhQhiT` R ^hiTI-=`4@TαhQT`BA`yV?C6D T[hQOiP->^OL= TT`VOQh¦R]2D jS^V OL]ghQT | ^&=^R V?>6=OQ` T6=^hi]7=`fh1F TUP_`TDw]ghQT@T``]!h<IVOi]!PS`E@TI `BA `VBCD T ;

8J$2')$78ng | ]PS` OL@->6=]PS`hQT`[A`VBC6D T hQOiP->^OL= T2x + 6y = 8

x P-;L: y2x + 6.00001y = 8.00001

x P-; y| T j-=]2]Sh<CDT T`yVBZtOQh!]gOQTUP R]!PK@OiV OQ]PgPK> d]2I-=zHJIg]O ZE>` ]2IQ@T= T hQT`[A`VBC6D T x P-;L: y Z x PK; y ;

| ]!PS`O@T>= ]!PS`D ^OiPVTUPS^P!VhQT`[A`VBCDT2x + 6y = 8

x P-; P y2x + 5.99999y = 8.00002

x P-; U yZV>` ]2IK@T=ThiT`BA`yV?C6D T~x P-; P y Zzx P-; U y ;Q| ]PSRUh<IS` OQ]P ;

8J$2')$78nL Y ]OiVAISPgTVD8^ V?= ORT[PS]P` OiP-\7IghQO<C= TV@GF ]2=z@T=T

n;5. ]PgPgT= IgP^h<\!]2=OVe-D TV@TV=?>` ]hLIV OQ]P~@T

A2x = bHJIgO >6R OVT hiTR^hRIghGT6f jShiOROiV Tu@T

A2 ; Rª^h<IST6=hQTN\!^OQPDTPPg]2D/]-= TN@T S]!jS`[` O A T`V[jghQTUOQPgT ; 8J$2')$78n$ | ]PS` OL@->6=]PS`hQT`ED8^ V?= ORT`

A =

4 1 1 11 3 0 01 0 2 01 0 0 1

, B =

1 0 0 10 2 0 10 0 3 11 1 1 4

XjgjghQOLHJIgT=hQ^/D >Veg]T@TN@GF >hiOLD OiP_^ V OQ]P@Tuv ^&I_` `Y^XRTRISPgT j-=B>RO`OQ]P @T10−3 ; N>6=O #ST=EH7IST­hQT`bÅ^!R V TI-=`

LU@T`~@T6I-f D8^ VB=ORT`8` ]PV ]2]VTUP9IS`8^XR!TR IgPgT]4]PgPgTj-=?>RUOQ` OQ]P>TUV HJIgThiT Pg]7Dw]-=T@T d @gT

BT`yV

D8^OiPV TP9I ; 8J$2')$78nq Y ]OiV

AhQ^ D8^ V?= ORTR^&=?=B>TM@GF ]2=z@T=T

n`[ISO<R ^PVT

A =

1 2 0 · · · 00 1 2 · · · 00 0 1 · · · 0;;; ;;; ;;; ; ; ; ;;;0 0 0 · · · 1

| ^hQR6IghQT6=detA ‖ A ‖1

TV ‖ A ‖∞;Q| ^hRIghQT6=

A−1 j-IgO` σ(A);

Page 44: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

U2U

Page 45: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

) !( ' (L

& # (

5,.+ 47?F0NM ;=>@?-:58 ?)4;5? -?A:5<-U@098b­P `F OQPVB>6=T`` TORO^&ITfnD >RU^PSOQ`BD T`u@TR^hRIghS@gT8hQ^j-=]! yTR VOi]!P ]7=Veg]2\!]PS^hiT @GF ISP RTRV TI-=u@T

Rn`BI-=NIgPI` ]2I_` ZwT` jS^RUT ; T`O=B>`[IShV^ V`Mb ]!PK@g^2DTPV^&I-f @GF TfO`yVTUPSRUT @GF IgPgOROiVB>TUVw@TRU^2=^!R V?>6=OQ`^ VOi]!P @T8h^jK= ]" yTR VOi]!P8]2= V eg]7\]!PS^hQTV@GF IgPDj4]OQPVY`BI-=IgP-` ]2IS`[ZwT` jS^RUT R ^h^&]ShiT`4@g^PS`@T`T` jS^RUT`jghLIS` \7>UP->=^2ITf~H7ISTRn `]!P!VE=^jgj4TUhL>`RUO ZÀ^jK=BC``^P_`@->6D ]PS`VB=^ VOi]!P ; #9Q@9jk8 =) !>@I"

L '>"IOI;oaH_@@a"%>@KU8

Rn "IC"N8>@!XGAB;=>@9C"y ∈Rn c 9"b

F.0;=>@9X"p

8L c #;F;='GU;C> O,a,".>@A>?"3rX>,lF>@*U

y

L c "bX88 ‖ y−p ‖≤‖ y−x ‖, ∀x ∈ L

0Xea>?\I".>@(CG,aOp'?9O@" KX"%n;=>8Dp ∈ L

>@I"U;C> O,a,".>@ >?"3rX>,lF>@*UDy

L

O#"y − p ∈ L⊥

uStuSt F<XF~ n 8 Fn~ 8 x |8|= x |= n!~ n~pr PgTt@T=]OiV THJIgO[jS^!` ` TjS^&= hF ]7= OL\OQPgTT`V~IgP `]7IS` ZwT` jS^!RT@T@O<D TPS`OQ]P :7; Y ]OiV

y ∈ Rn TVDhQT` ]2I_` ZwT` jS^RUT@TNR!TRV T6IK=`YTP-\TPK@T=?>`jS^&=EISPoR!TRV T6IK=

vPg]PP9Igh %

D = z ∈ Rn/z = xv, x ∈ R ;Y O

p ∈ DT`V­h^8j-=]! yTR VOi]!PK]7=Veg]2\!]PS^hiTN@T

y`BI-=­hQ^ @-= ]!OVT ]!P j_TIVO>R=OL= T&% y = p + u ^XRTR p = xvTUV

uT v = 0;b­PN=T6D8^2=?HJIgTSHJIgTh^V@T= PSO<C= T =TUh^ VOi]!P ` OL\PgO$#ST HJIgT

u ∈ D⊥ TV#@]!PSRSHJIgT uT`V»h^j-=]! yTR V OQ]P ]7=Veg]2\!]PS^hiT

@gTy`[IK=

D⊥ ; x OL\2IK= T UK;L: y ; TRU^hQR6Igh @T pT`yV[^hQ]2=`

vT y = xvT v ⇒ x =1

vT v.vT y

` ]OiVp =

vT y

vT v.vXOiPS` O RTV V TYTfj-=T`` Oi]!PuH7ISOTfj-=O<D ThiT4bÅ^OiVHJIgT p

^h^D6,D T@OL=TR VOi]!P/HJIgTvj_TIV`6F >R6= OL= T@O*>= TD/Z

D TP!V j4]2I-=M= Tj-=?>` TUPV T=[hQ^VB=^PS`[b ]2=?D8^ VOi]!PtH7ISOV?=^PS` b ]7=BD TyTP

p%4]PtR7>6=O #_T/HJIgT

(vT y).v = (vvT )y]mhF ]!Pm]2]S` T6=?RTuHJIgTvT y

T`VMIgP ` R^h^OL= T^hi]7=`H7ISTvv⊥

T`yVOIgPgT/D8^ VB=ORT @T =^P-\ : xÅj-IgO`BHJIgTcV ]2IgV T`hQT`DR]hQ]PSPgT`` ]PV @T` DwIShVOijghQT` @Tvy~HJIgO +% (8J6(8 hF T`jS^!RT

Rn `BI-=h^ @T= ]!OVTD; »^Fj-= ]" yTRV OQ]P]7=Veg]2\!]PS^hiTY`BI-= IgPgTV@T=]OiV TVHJIgOSjS^``TYjS^2=hF ]7= OL\OQPgTYT`V @]P_RIgPgTYVB=^P_` b ]7=BD8^ VOi]!PchQOiPK>^OL=T @TVD8^VB=OQRUT1%

P =1

‖ v ‖2 vvT

U9W

Page 46: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

U7^ 1 1

UK;L: dS= ]" yTRV OQ]P`[I-=EISPgTN@T= ]!OVTi 89jk+-XG"%89p %

:7;| ]2D D Tp = Py h^cj-=]! yTR VOi]!P]2= V eg]7\]!PS^hQT`BI-=YhiT `]7IS`[ZtT`jS^!RT­]2= V eg]7\]P_^h D⊥ T`yV

u = y− p =(I − P )y @]P_R I − P

T`VhQ^ D8^ V?= ORTN@Tj-=]! yTR VOi]!PD]2= V eS]2\]!PS^hQT `[I-=D⊥ ;

-;VY ]OiVθh1F ^P-\!hiTTUPV?= T hQT`V@OL=TR VOi]!PS`E@T`RTRV TI-=`

vTUV

y; b­PK^ ^hQ]2=`1%

cos(θ) =vT y

‖ v ‖‖ y ‖TV[]PKTUP@T>(@TIgOiV[h1F OQP->6\^hQOV?>O@gT Y Re ^2= %

(∀a, b)|aT b| ≤‖ a ‖‖ b ‖

8Jj +%018 %v =

[

10

]

| ]2D D TvT v

: ]P ^ ]gOQTUP p = Py =

[

y1

0

]

uSt6u3v F:0X~Sn D !n~% x |8:| x |yxD| 6!~ n~S

b­PNIVOihQOQ` Tfh^V=TUjK=B>`TP!V^ VOi]!PM@GF IgPgT@T=]OiV TfRU]2D D T IgPc` ]2I_` ZwT` jS^RUTq^ 8PgTfj_^&=^hQh<ChiT 9EISPc`]7IS`[ZtT`jS^!RT%D = z ∈ R

n/z = z0 + xv, x ∈ R

Y ]!OVy ∈ Rn TV p

`^ljK= ]" yTR VOi]!P-]7=Veg]2\!]PS^hiT`[IK=hQ^ @T=]OiV TD; Xhi]7=` %

y = p + u^XR!TR

p = z0 + xvTUV

uT v = 0 @GF ] p = z0 + P (y − z0) = (I − P )z0 + Py ]PT`yVhQ^tD ^VB=OQRUT @TjK= ]" yTR VOi]!PI`[IK=chQTD`]7IS`[ZtT`jS^!RT TUPK\TUPQ@T=B>DjS^&=

v]2]gV TUP9Ik@g^PS`hQTjS^2=^2\2=^jgeST

UK;L:2;L:2;

uSt6u F:0X~Sn D n ,0:xD|C

Y ]!OVAISPgT/D8^VB=OQRUT

(n × r)@T/=^PK\

rx)@g]PSR

r ≤ ny ; | ]P_`O@T>6=]P_`[h^j-=]! yTR V OQ]P

p@GF IgP RTRV TI-=

@TRn `[IK=hiT` ]2IS`[ZwT` jS^RUTOLD ^2\TN@T A

x OL\ UK; y ;

Page 47: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

UT_

UK; dS= ]" yTRV OQ]P`[I-=EISP` ]2I_` ZwT` jS^RUT

Xhi]7=`y = p + u

^XR!TRp = Ax, x ∈ Rr TV AT u = 0

R^&=u ∈ Ker(AT )

;b­Pl]2]VOiTPV^hi]7=`%AT y = AT Ax TVRU]2D D T A

T`V@T=^P-\jghiTOiP hQ^D8^ VB=ORTx r×ryAT A

T`yVqOQP9RT6=` O<]ghQTTUV %p = Py

^XRTRP = A(AT A)−1AT ;

. TjghLIS` ]Po= TUVB=]2I-R!T­h^ D8^ VB=ORTN@Tj-=]! yTR V OQ]P-]2= V eg]7\]!PS^hQT­`BI-=hQTPg]A!^2I @T AT %P ′ = I − P

;

uStu |!~SC< x F<XF~ n

T`ED8^ V?= ORT`@Tj-=]! yTRV OQ]PK`]!P!VV@T`ED8^ V?= ORT`EH7ISOGj_]` `BC@TPVYhQT`@T6ITf-j-= ]!j-=O<>UVB>`Y`BIgOLRª^PV T` %

P = P TP 2 = Pb­PoR2>= O$#ST=^RT`Yj-= ]!j-=O<>UVB>`Y`BI-=hQT`ED8^ V?= ORT`j-=B>R>(@TUPV T` ;T`ND8^VB=OQRUT`N@T8j-=]! yTR VOi]!PF` ]PVTUP \7>UP->=^h`OQP-\2IShiOLC6=T` j-IgO`BHJI!F TUhQhQT`N=^&D CUPSTUPVh1F T` jS^!RT&9IgP `]7IS`[ZT`j_^RTN@TN@OLD TUPS` OQ]PjghLIS`Yj_TUV OiV T ;. Tjgh<I_` ThihQT`RU]PVB=^RV TP!VYhQT`Pg]7=BD T`% ‖ Py ‖≤‖ y ‖ ;

5,KJ ?A>.:5C54I098 Mu<)4I47698 U@>@:B69<A>V47098

u3vfuSt < ~C|F~ n 0yx |=F| \F<

r PI`[A`VBC6D T T`V]2]S` T6=?R2>09KjS^&= V OL=u@T`ND T`BI-= T`O@T` ^` ]2= V OQTyTUP b ]P_R V OQ]P @T @O>6=TUPV T` TUPVB=?>UT`

t; b­PRTIVR ^hiO@T6=IgPtD ]T@TCUhQTVe->U]7= OHJIgTw@TRTc`BA`yV?C6D T/HJIgO%@T>Uj4TUPK@ @T

njS^2=^2D CVB=T`

x1 · · ·xn;Y ]OiV

x = (x1 · · ·xn)T hQT4R!TRV T6IK=@gTj_^&=^&D CV?= T` 9M@T>VT6=?D OiPgT= ; b­P ^@g]PSRj_]7I-=ReS^7H7ISTfTUPV?=B>T ti, 1 ≤ i ≤ mISPgT`]7=VOiTMD T`BI-=?>UTYiTVVIgPgT` ]2= V OQTV e->]2=OLHJIgT

yi = f(ti; x)x O<\ UQ; P y

Page 48: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

U7 1 1

UQ; P f(t) = x1 + x2t + x3t

2 D T`BI-=?>UT j_]7I-= t = t1 · · · t5SF T6=?=T6I-=8TP!V?= T hQTD ]T@TCUhQTV e->]2=OLHJIgT TUVDh^ `]7=VOiTD T`BI-=?>UT T`V

ei = yi − YiTUVhQ^ ` ]hLIVOi]!P ^2ITf

D ]OQPK@T=T`R^&=?=B>`RU]PS` O`yVT 9 Reg]!OQ` OL=xHJIgO DOQPgOLD OQ` ThQ^ RU]2D D TO@gT`[R^&=?=B>`4@T`T6=?=T6I-=`1%

` = ]7I-RT=x ∈ Rn V Th HJIgT m

i=1

e2i

` ]OiVEDOQPgOLD8^hQT ;Y O¦h^Ob ]PSRV OQ]P

fT`yV 0)'1#9+-')X8 +%+-/+ ++ Q( +-" ++T+-jyJX89p

x ]PjS^2= hQT@Tj-=]2]ghLC6D T^2I D ]OQPK@T=T`RU^2=B=?>`chiOQP->^OL= T` ; OLDOiV ]!PS`lPg]2IS`69 RUT-R^`lTVlOQPV T= j-=?>V]PS`lhQTDj-=]2]ghLC6D To@g^PS`ch1F T` jS^!RT @gT`/DT`[IK= T` %ei = yi − Yi = aT

i x− Yi;

Y ]!OVehQToR!TRV T6IK= @gT

Rm @]!PV hQT` RU]2D j_]` ^PV T`` ]PVhQT` T=B=T6IK=`eiTUV ` ]OiV

AhQ^ D8^ VB=ORT x

m × ny

@]!P!V[hQT`YhQOL\PgT``]!P!VhQT`aT

i

; Tj-=]2]ghLC6D TR]!PS` OQ`V T^hQ]2=` 9` = ]7I-RT=

x ∈ Rn H7ISO jk')%')j ')p(8 ‖ Ax− Y ‖2 ;

. ^PS`h1F T` jS^RUTRm Oih `6F ^&\!OV@g]PSRu@gTVB=]2I-R!T6=[hQTj4]OQPV@ThF OLD8^&\!Tu@T A

hQTjghLIS`[jK= ]RegT^2Im`TPS`@Th^ Pg]2=?D T8T6ISRUhiO@OQTUPgPST @TI R!TR VT6I-=Yx)HJIgOY^j_TI @TDRe_^PSRUT`u@GF ^jSjS^&= V TPgO<= 9

Im(A) RU^&= m >> ny ;SF IgPgOHJIgT8`]!h<IgV OQ]P @g^PS`hF T`j_^RT @T`ODT`[IK= T`­T`yVN@]!PSRhQ^ +# (8J$2'1Q QX%*Q%+-018 @TI R!TRV T6IK=Y`BI-=hiT` ]2I_` ZwT` jS^RUT

Im(A); ^ ` ]hLIV OQ]P^~@T> 9 >UVB>RU^hQR6IghL>UT^&IKjS^&=^&\7=^jgegT UK;L:2; P R&F T`VhQ^ ` ]hLIV OQ]P@TI`BA `VBCD ThiOQP->^O<=T `[IgOLR ^PV @OV^2ITf 9G"%+T'1Q%p #Qjk+-0)89p %

AT Ax = AT Y

Y OrangA = n

x e9A j_]V e-C`Tw=^OQ` ]PgP_^&]ghQTRU^2=m >> n

TV­hQT`ai@T>Uj4TUPK@gTUPVM@T`TPVB=?>UT`

tiy RUTc`[A`VBCDT^tISPgT-` ]hLIV OQ]P IgPgOH7IST

x = (AT A)−1AT Y @OiV T-` ]hLIVOi]!P>^&ITf D ]OQPK@T=T`RU^2=B=?>` ; ^nD8^ VB=ORT A+ =(AT A)−1AT x XVV TPV OQ]P qRUTDP!F T`VljS^!`ch^tD8^VB=OQRUT @T-j-= ]" yTRV OQ]PQyT`V` ]2I-R!TUPVl^jSj_Th<>T-h^ +p89"% #') 7*8JXp(8 @gThQ^ D8^ V?= ORTO=TR V^P-\7Igh^OL= T

A; ohQhQTc` ^V O` bÅ^OV

A+A = ITUV

AA+ = P;

u3vNu3v : 0y~SDC x |F~DS<Y ]!OVIgPC`BA`yV?C6D T hQOQP->^O<=T OQPSR]7D jS^ VO<]ghQT

Ax = b ] A(m × n)T`VKV TUhQhQT HJIgT

rangA < mTV

b /∈ Im(A); b­PI`BIgjgj4]!` T6=^KjS^&=T6fT6D jghQTnxÅRU]2D D T @g^P_`hiT-RU^`w@T`ND ]!OiPK@-= T`RU^2=B=?>`zyOHJIgT

m > nTUV

rangA = n; T`[A`VBCDTcP!F ^ @]P_RjS^`M@T`]!h<IVOi]!P TUV­]!PmhiT/= TD jghQ^!RTj_^&=hQTl`BA`yV?C6D Tc^2ITf>HJIS^ VOi]!PS`Pg]7=BD8^hQT`Y]7]V TP9I-TPKhQTODwIghiV OQjghQOQ^PV[jS^&=

AT %

AT Ax = AT b

Page 49: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

1 UJc| T `BA `VBCD T­T`yVTUP \2>P->6=^h*D8^hGR]!PK@OiV OQ]PgP->R^&=hQ^R]!PK@OiV OQ]P @gT

AT AT`VYhQ^R]!PK@OiV OQ]P@T

A^&IKRU^2=B=?>

@S^PS`qhQTR^`S@GF IgPgTD ^VB=OQRUTRU^2=B=?>UTA; ^OD >UV eg]T@T@gTv ^&IS`` =OQ`?HJIgTV@GF ,V?= T[OQPgT8R^RUTTVf]!P8h<IgO_j-=?>b>6=T6=^@gT`D >Veg]T@T`%]S^!`[>T``[IK=@gT` VB=^P_` b ]7=BD8^ VOi]!PS`»]2= V eg]7\]P_^hQT` HJIgO]!PV hF ^XR ^P!V^&\!T4@GF ,V?= TYP9I-D >6=OLHJIgTD TUPV`V^2]ghQT` ;

OQPS^hQT6D TP!V `OGhQT`YRU]hQ]PgPgT`@T A` ]PVY]2= V eg]!Pg]2=?D >UT`x ]_~aZ ~ ]2= V eS]2\]!PS^hQT`4@T6I-f9/@gT6ITfDTUVV@T Pg]2=?D T : y

AT A = IxÅh1F O@TUPV OiVB>N@g^PS`hQ^ F j4TVOVT2F-@OLD TUPS` OQ]P

nyfTVYhQ^` ]hLIV OQ]P @gT`>HJIS^V OQ]PS`fPg]2=?D8^hQT`T`VY` O<D jghQTZ

D TP!Vx∗ = AT b

; »^tF ]_]!PgPgT7F¦`VB=^ V?>6\!OiTlj_]7I-==?>` ]2IQ@T= TlhiTj-=]2]ghLC6D T @T`MD ]OQPK@T=T`RU^2=B=?>`T`VO@]!PSRw@gTRU]PS`VB=?IgOL= TIgPgTO]S^!`T]2= V eg]!Pg]2=?D >UT@gTIm(A)

j4]2I-=[R^hRIghQT6=TfjghQOQRUOVT6D TUPVh^cj-=]! yTR VOi]!P ; b­P RT=B=^c^2Ij_^&=^&\2=^jSegT UK; P HJIgTlRUTVVTRU]PS`VB=?ISRV OQ]P=T6R OiTPV98V?= O^P-\7Igh^&=OQ` T6=h^ D8^ V?= ORTcjS^&=O@T`[VB=^PS`[b ]2=?D8^ VOi]!PS`]7=Veg]2\!]PS^hiT` ; u3vfu f x !z6F<88~ p~S zo|~ F

Y ]!OVIgP D ]T@TCUhQT^ PST@GF IgPD`BA `VBCD T9NIgPgT­TUPVB=?>UTt%y = a+ bt ] a

TVb` ]PV4@gT6ITf jS^&=^&D CUVB=T` 9

@->V T=BD OQPgT6=j4]2I-=D OiPgOLD OQ` T6=Y^&I-`TPS`4@gT`4D]!OiPQ@T= T`fR^&=?=B>`hF T=B=T6I-=@T`fVB=]O`SD T`[I-=T`Mxt, Y

yf`[IgOLR ^PV T`%xZ : U y x d W y x : c y ;T`[A`VBC6D T

(S) :

a − b = 4a = 5a + b = 9P F ^ ]gOQTUP `-=jS^`E@gT`]!h<IVOi]!P ; b­PR]P_`yV?=BIgOiVV@]P_R hiT`>HJIS^ VOi]!PS`YPg]7=BD8^hQT`TP D/IghVOijShiO^PV[jS^2=

AT =

[

1 1 1−1 0 1

]

TV@]!PSR AT A =

[

3 00 2

]

@]P_R (AT A)−1 =

[

1/3 00 1/2

]

@!F ]-h^8`]!h<IgV OQ]P^&ITf D ]!OiPK@-= T`R^&=?=B>`a = 6 =

TVb = 5/2

;b­P = TD8^&=zH7ISTHJIgT hiTMRTR VT6I-=E@T`T=B=T6IK=`Y −Ax = [1/2 −1 1/2]T

T`yV]gOQTUPK]2= V eg]7\]!PS^h*@g^PS`R3 ^&ITfRU]hQ]PgPST`u@T

A;S| eS^2HJIgT8T=B=T6I-=cj_TIV ,V?= T >6\!^hiTD TUPVu= Tj-=?>` TUPVB>T8jS^&=h^t@O`V^P_RT~RT=VOQR^hQT8TUPVB=Th^D T`[I-=T

YiTUV[hQ^ @T=]OiV T

f(t) = 6 + 52 tj4]2I-=

t = tix O<\ UQ; U y

UK; U ~=?>6\7= T` ` OQ]PhQOiPK>^OL=T5, O47<A:B8v?)4 JL<T;=>@?-:58 ?)4;5? -?A:5<-U@098 ufuSt |!~SC<y6 nn| 0 4%%'(')* 1 0XD& ".K3aea'pOG#

HO#"f:3"% KX% Q*#+K0)8 p

HT H = HHT = I

r PgToD ^VB=OQRUTD]7=Veg]2\!]PS^hiT8T`yV~@]P_R IgPgToD8^ V?= ORTKRU^&=?=?>UT @]!PVlhQT` R]!hi]!PgPgT`l`]!PVl]2= V eS]Pg]7=BD >UT` ; T`D8^VB=OQRUT`@gT/=]V^ V OQ]P @T `[A9D >V?= OQT @Tlj_T=BD/IV^V OQ]PmTUVhF O@TP!VOV?>` ]PV@gT`TfT6D jghQT`M@T/D ^VB=OQRUT`­]2=BZ

Page 50: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W&d 1 1

V eS]2\]!PS^hQT` ;r PSTD8^ V?= ORT­]2= V eg]7\]!PS^hQT

HT`yVPS^ V?I-= ThihQT6D TP!VYOQPJR!T6=`OL]ghQT[jS^2=@T> #SPgOiV OQ]P TVYhF OQP9RT6=` T[T`V H−1 = HT ;

+4Qp('(')* =) 6O>O;CKG@"bG g3>?\?&(X"b?O6"3'!Sg3>?p& F".>@e>@,".rX>@l>@X?C'>?C"C8\k>@&(G@"3p c oC>@K& OqU?ap9.:UFXSs c !;CO>F!I"30Oa?U?9O@"D?lUO>@C"a>?!',_@G

H>@,".rX>@l>@X? ⇔ (∀x ∈ R

n) ‖ Hx ‖=‖ x ‖oPT ¦TV ‖ Hx ‖2= (Hx)T (Hx) = xT HT Hx = xT x =‖ x ‖2 ;

| TV V T8j-=]jK= OL>V?>lTP!V?=^ PST ISPgT8`yV^&]gOQhiOiVB>8P9I-D >= OHJIgT~@T`OD >Veg]T@T`OIgV OQhiO` ^PVcRT`V?=^PS`[b ]2=?D ^V OQ]PS` ; b­PhQT`EIVOihQO`Tj-=OQPSROQjS^hQT6D TP!Vj4]2I-=:% ]2= V eg]!Pg]2=?D8^hQOQ` T6=IgP`[A`VBC6D Tu@TO\7>UP->=^V T6IK=` =B>`]7IK@T=TIgP `BA`yV?C6D T`^2ITf >HJIS^V OQ]PS`Pg]7=BD8^hiT` VB=OQ^P-\2Igh^&=O`T=4IgP `BA`yV?C6D TOD ^hR]!PK@OiV OQ]PgP->RU^hRIShiT=hiT`Rª^hiTI-=`Yj-= ]!j-=T`@GF IgPgTND8^ V?= ORT xÅReS^jgOiVB=T W yuNu3v wF 6n |S~S|F~ n F| f ~

XCjS^&= V OL=@!F IgPSTbÅ^2DOQhQhiTO@TMRTRV TI-=`OQPK@T>Uj4TUPQ@g^PV`@TRp ]7I @GF IgPgTMD8^ V?= ORT A ]PDj_TIV[R]P_`yV?=BIgOL=TIgPgT bÅ^&D OQhihQT q1 · · · qp ]S^` TD]7=Veg]PS]2=?D >T @T Im(A)

; | F T`yV IgP>]7IV OQhb ]PK@S^&D TUPV^hj4]2IK=lh^t=?>` ]hLITZV OQ]P @T `[A`VBCDT` `[I-=z@T>UV T6=?D OiPK>` ; F O@T>UT \7>UP->=^hiTT`yVu@]!PSR @TRU]PS`VB=?IgOL= T/IgPgT ]S^` T]2= V eg]!Pg]2=?D >UTw@TI` ]2IS`[ZtT`j_^RTO<D8^&\!T@GF IgP-TUPS` T6D/]ghiTO@TRTRV TI-=` ; SF OiPVB>=-,UVP9I-D >= OHJIgTT`yVEHJIgT­RUTVVTRU]PS`VB=?ISRV OQ]P >(H7ISO ZR ^&IV9cVB=OQ^P-\2IShQ^2= O`T=fh^wD8^VB=OQRUTVb ]2=?D >UTjS^&=YRT`RTRV TI-=` ; b­PDj-=?>` TUPVT6=^OQRUOGhQ^R]!PS`VB=?ISR VOi]!P @T RTV V T]S^!`T[]7=Veg]PS]2=?D >T RU]PgP9IgT[` ]2IS`hiT[PS]2D @Tj-=] R6>@TIK= TV@TMvM=^2D/Z Y Re-D O@ V @]PVhF OQPVB>=-,UVfT`Vfj-I-=T6D TP!V^R^2@T>D OLHJIgT ; oPj-=^V OHJIgT vM=^&D Z Y Re-D OL@VRU]VTVB=]jRegT6=TUVY]!P IgV OQhiO`Th^ D >V eS]9@gTu@TMbÅ^R V]2=OQ`^ VOi]!P i HJIgOgj4T6=?D TVS@GF ^ V V TUOQPK@T=ThQTD0,DT]7] yTRV O<bQ\7=!RT 9M@gT` VB=^P_` b ]7=BD8^ VOi]!PS`»]2= V eg]7\]P_^hQT`#>UhL>6D TUPV^O<=T`x Q+T(')*#pO %8 mo'87*8J%p ]2IlVB=^P_` b ]7=BD8^ VOi]!PS`%@T Q"%p(8 %*0) %87 yP9I-D >6=OLHJIgTDTPVq`V^2]ghQT` TVf` T6IghQT6D TP!V@TITf b ]!OQ`YjghLIS`[ReKC6=T`HJIgThQ^ D >Veg]T@Tu@Twv ^2IS` ` ;| ]P_`O@T>6=]P_`fV]2IV@!F ^2]_]7=?@ IgPgTND8^ V?= ORTR^&=?=B>T

A ∈Mn,n(R)@]!PV[hiT`R]hQ]PSPgT`

ai` ]PVYhiOQP->^O<=T6D TP!VOQPK@T>Uj4TUPQ@g^PV T` ; k ]PV?= ]!PS`EHJIgTh1F ]Pj4T6IVR]P_`yV?=BIgOL=TOiVB>=^V OLRTDTPVISPgTuD8^ V?= ORT ]2= V eS]2\]!PS^hQT

Q @]PVhQT`[RU]hQ]PgPgT`Y`]!P!VPg]VB>T`qiV TUhQhQTOHJIgT

QT A = R ] RT`yVYV?= O^P-\7Igh^OL= T `[ISjQ>= OQT6I-=T ;

| ]2D D TQT`V[]2= V eg]7\]P_^hQT ]PK^ % A = QR

; b= hQT`>h<>D TUPV`E@T R`]!P!VE@Th^/b ]2=?DT0%

r11 = qt1a1

r12 = qt1a2, r22 = qt

2a2;;;r1n = qt

1an, · · · rnn = qtnan

b­PKOL@TPV O$#ST^hQ]2=`hiTj-=]T@TIgOiVQR

jS^&=[RU]hQ]PgPST6%

a1 = (qt1a1)q1 ⇒ q1 = a1

‖a1‖TV

r11 =‖ a1 ‖a2 = (qt

1a2)q1 + (qt2a2)q2, ⇒ q2 =

a2−(qT1

a2)q1

‖a2−(qT1

a2)q1‖;;;b­P= TD8^&=zH7ISTMHJIgTReS^7HJIgTRU]hQ]PgPSTqiT`V[]2]gV TUP9IgT TP` ]2I_`yV?=^XA^PV@T

qi`T`Yj-=]! yTRV OQ]PS`]7=Veg]2\!]PS^hiT`

Page 51: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

1 W-:`BI-= hQT`

i − 1j-=T6D OQT6=`R!TRV T6IK=`M@ThQ^ ]S^` T]2= V eg]!Pg]2=?D >UTw@T> 9-R^hRIghL>` j-IgO`TUP PS]2=?D ^PVhQT/=?>`BIghiV^Vx O<\ UQ; W y

UQ; W vM=^2D Z Y ReKDO@ V

b­PFj4T6IVu=?>^hiO`T= hQ^D0,6D T8R]!PS`VB=?ISR VOi]!P´` OhQTPS]2Dw]K= T~@T~R!TR VT6I-=`aiT`VOiP-b>6=OiTI-= 9

nx)D ^VB=OQRUT

=TRV^P-\2Igh^OL=Twxn× p

yBy ; b­PDRU]PS`VB=?IgOiVpR]hQ]PSPgT`

qi]7=Veg]Pg]7=BD >T`RU]2D D T­j-=?>R6>@TD DTPV@TMD8^PgOLC6=T

9]7]V TPgO<=A = Q1R1 ] R1

T`VVIgPgTOD8^ VB=ORTRU^2=B=?>UT­VB=OQ^P-\2IShQ^O<=T xp× p

y ; b­PR]7D jgh<CUV T^hi]7=`h^/]_^` T]7=Veg]PS]2=?D >TlTP R]PVOiP9IS^P!Vh^j-=] R6>@TIK= T~@T vM=^&D Z Y Re-D OL@Vc^XRTRn − p

RTRV TI-=`^2=B]SOV?=^O<=T` D ^OQ`VTUh`OH7ISThQT`nRU]hQ]PgPST`b ]2=?D >T` ^XRTRhQT`

ai`]!OiTP!V hQOiP->^OL= TD TUPVOQPK@T>j_TPK@g^P!VT` ;!Y ]OiV

Q2h^ D ^VB=OQRUT

@gT`n− p

@T= PgOQT6=`RTR VT6I-=`]2= V eg]!Pg]2=?D >` ; b­P^ ^hQ]2=`]gOiTP %AT Q2 = RT

1 QT1 Q2 = 0

RUTOHJIgO D]!PVB=TOHJIgT0%A = QR = [Q1Q2] =

[

R1

0

]

= Q1R1;

8Jp $7Q01Q%#89p #8Q1

Kjk89 "#%8%+Kp8QX%Q%Kjk98o #8Im(A)

8J 0189p/$7Q01Q%#89p #8Q2 Qjk89 "%#8+-p(8tQ(#*%Kjk98 #8

Ker(AT )

ufu AF| '6 |~Sn <

b­P-j_TIVOQPV T= j-=?>VT6=hQ^wD >UV eg]T@TO@TOvM=^2D/Z Y Re-D O@ VRU]2D D TIgPSTD >Veg]T@TO@T[V?= O^P-\7Igh^&=OQ`^ VOi]!P @gTh^ D8^ VB=ORTA ^&IoD6,D T V OiVB=TuH7ISThQ^ D >Veg]T@Tu@Twv ^&I_` ` ;T hT`V[j4]!``OL]ghQTO@gTN=B>]2=?\!^PSOQ` T6=hiT`RU^hQR6IghQ`YTPRU]PS`VB=?IgO` ^PVw@T`VB=^PS`[b ]2=?D8^ V OQ]P_`O>UhL>6D TUPV^O<=T` ]7=Veg]2\!]PS^hiT`NHJIgOfT4TR VBISTUPVlRTUVVTVB=O^P-\7IghQ^2= O`^ V OQ]PRU]hQ]PgPSTj_^&=cRU]hQ]PgPgTxÅ]2I >UhL>6D TP!Vj_^&=N>UhL>6D TP!V y ; T`c`[A9D >V?= OQT`N@T-n[]2I_`Teg]h@T6=TUVhiT`N=]V^V OQ]PS`N@gTvOLRTPS``]!P!VO@T`T6f TD jghiT`­` OLDjShiT`TUVOQPVB>6=T``^PV`@TcVTUhQhiT`­VB=^PS`[b ]2=?D8^ V OQ]P_` R^&=TUhQhiT`­RU]PK@-IgOQ` TUPV:9@gT`^hL\]7= OiV eKDT`PJIKD >= OHJIgT6D TP!VjghLIS`[`V^2]ghQT`EHJIgTh^/D >UV eg]T@Tu@TuvM=^&D Z Y Re-D OL@V ;

4%%'(')* g 0X & F".K.aOd d>8'rX>?.'"XN& F".K.aOda?KGH

o87 G,apKI"H = I − 2P c >$#

P"k`& "3p3ae8f;\> O,a"3U>?BNUB>?3"%d'lF'\OG# ;om_@,a,"b?

vX>@jC

b­PoR2>= O$#ST~x OL\ ;-UQ; ^ yH7ISTH=TUj-=?>` TUPVTIgPST `BA9D >VB=OQTj_^&=E=^jgj4]2= V[^&I`]7IS` ZwT` jS^!RT

v⊥;

Page 52: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W 1 1

UK; ^ ` =^PS`[b ]2=?D8^ VOi]!P @Tn[]7IS` TUeg]!hL@T=d]2I-=fVB=OQ^P-\2IShQ^2= O`T=qhQ^cR]hQ]PSPgT

k@GF IgPgTVD8^VB=OQRUT

A ]P8T ¦TRVBIgT=^hQ^ V?=^PS`[b ]2=?D ^V OQ]P8` ]2IS`qh^u@O^&\!]&ZPS^hiT j4]2I-=R]P_`T=BR!T6=qhQT`S>h<>D TUPV`qP9Igh`fRU]PS`VB=?IgOiV`^&ITfOV?>6=^ V OQ]P_`qj-=?>R>(@TUPVT` ;JY ]!OV akhQT` ]2IS`[ZR!TRV T6IK=^`` ] RUO<>l^&ITf

n− k + 1@T6=PgOLC6=T`­R]2D j4]!`^PV T`V@TlhQ^-R]hQ]PSPgT

k@T

A; b­Pn@T>VT6=?D OiPgTlhQ^D`[A9D >V?= OQT

Hk@g^PS`Rn−k+1 V ThihQTuHJIgT Hkak = zk ] %

z1k =‖ ak ‖zik = 0, 2 ≤ i ≤ n− k + 1oPT ¦TV Hk

T`VY]7=Veg]2\!]PS^hiT RTNHJIgOOLDjShiOHJIgTuHJIgT ‖ zk ‖=‖ ak ‖; b­PK]2]VOiTPV^hQ]2=`%

Hk = I − 2vkvT

k

vTk vk^XRTR

vk = ak± ‖ ak ‖ e1 TV e1T`yVhQTj-= TD OiT=ERTRV TI-=E@ThQ^ ]S^!`TR^Pg]!PgOLHJIgTN@T

Rn−k+1 ;oP j-=^ VOLHJIgT ]PReg]O` O<=^ vk@TPg]7=BD TOD8^fOLD ^hiT ;

dhLIS`#\2>UPK>6=^hQT6D TUPV hiTfV e->]2=?C6D T`BIgOLR ^PV D ]PVB=T4HJI!F OQh!T`yV»V ]2I! y]2I-=`j4]!``OL]ghQT4@TfVB=]2I-R!T6=!ISPgT D8^ V?= ORT@Tn[]2I_`Teg]h@T6=j_T=BD TV V^P!VM@TVB=^P_` b ]7=BD T6=akTP

zk TUVjghLIS`\7>UP->=^hiTD TUPVVIgPRTR VT6I-=MH7ISTUhR]PQH7ISTTUPIgPR!TR VT6I-=[RU]hQOiPK>^OL=T19 IgPRTR VT6I-=V@]!PgP-> ; %9K?9jk8 g !>@'C"

f@"

e@ _?,a"%@'`X>@ a>@99XGOO 8

Rn n_?,a ‖e‖2 = 1 % "dU>?K

;=>?KSb^?H8H".O>8F_?'u ∈ Rn "%'

‖u‖2 = 1 H(u)f = αe

g Z[T6D8^&=zHJIg]PS`»V]2IVS@GF ^&]4]2=z@/HJIgT` OH(u)

T`V IgPgTED8^ V?= ORTE@Tn[]7IS` TUeg]!hL@T= ^hi]7=`H(u)f = f − 2u(uT f)

TV ‖H(u)f‖2 = ‖f‖2;d]!` ]PS`^hi]7=` | α |= ‖f‖2 ; b­PRegT6=RegT ^hQ]2=` uV TUh!HJIgT

H(u)f = αe ` ]OiVf − 2u(uT f) = αe

u =1

2uT f(f − αe)

Y Oβ = uT f TUPDwIShVOijghQO^PV9 \!^&I_RegT­j_^&= fT %

2β2 = α2 − αfT eTVβT6f O`V T` O

α2 − αfT e > 0; b=hF OQP->\!^hQOiVB>N@T | ^&I_ReJAJZ Y Re ^2= Pg]2IS`E@g]PgPgT

| fT e |≤ ‖f‖2‖e‖2 = ‖α‖

Page 53: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W2PTUV[h1F OQP->6\^hQOV?>T`yVV@TjSh<IS``yV?= OR VTj_^&=e9A j_]V e-C`T~x

fTV

ePS]PR]hQOQP->^O<=T`zy ; X[OQPS` O %

u =1

2β(f − αe)

=?>Uj4]PQ@9h^~H7IST`V OQ]P ;i 89jk+TG"#8 =) 8

f@"

e>@C"a>?9XGOO c H = I

>8H = I − 2eeT OG ;=>@o8'C" k(O#"3U>?

»^6#K\2I-=T UK; _ jK=B>`TP!VTOIgPTfT6D jghQTN@TR]!PS`yV?=BI_R V OQ]Po@g^P_`R2 ;

UQ;e_ ` =^P_` b ]7=BD8^ VOi]!P @Tn]2IS` TUeS]h@T6=b]S` T6=?R]!PS`wHJIgTKh^mV?=^PS` b ]7=BD8^V OQ]P ]2= V eS]2\]!PS^hQTD^!` ` ]ROL>UT 9 hQ^ VB=O^P-\7IghQ^2= O`^ V OQ]P @TKh^ RU]hQ]PgPgT

k@S^PS`Rn T`Vh^ND8^ V?= ORT Qk

= Tj-=?>` TUPVB>TRO<Z @gT``]7IS` ]_~aZ ~ hQ^wD ^VB=OQRUT[O@TUPV OiVB>O@T Rn ]hQT]ghQ]R@O^&\!]PS^h@gT`n− k + 1

@T= PSO<C= T`RU]hQ]PgPST`^ >V?>N= TDjShQ^!R> jS^&=Hk;

Q =

1 ; ; ;1

Hk

5, 034=Mu>#Mu098 8J$2')$78 =) ` =]2I-R!T6=Yh^/D TOihQhiTI-=T^jgj-=]XfO<D8^V OQ]P x ^2IK` TUP_`E@T`D ]OQPK@T=T`R^&=?=B>` @TI `BA`yV?C6D T

3x = 10, 4x = 5.

. >VT6=?D OiPgT=fhQTRU^2=B=?>E@ThF T6=?= TI-=e2 TV4D]!PVB=T6= HJIgT[hQTVR!TR VT6I-=fT6=?= TI-= (10−3x 5−4x)T T`yV]2= V eg]7\]P_^h

9(3 4)T

8J$2')$78 =g ` =]2I-R!T6=hQ^/@T= ]!OVTf(t) = at + b

H7ISO¦^jgj-=]RegThiTMD OiTITftxÅ^2ID`TPS`@T`4D ]OQPK@T=T`fR^&=?=B>`?yhQT`D T`[I-=T` %f(0) = 0, f(1) = 1, f(3) = 2, f(4) = 5.

Page 54: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

WU 1 1

87$7')$28 Y ]OiVP =

1

‖ v ‖2 vvT ,

h^ D ^VB=OQRUTN@T j-= ]" yTRV OQ]P`[I-=h^ @T=]OiV TN@Tu@OL= TR VOi]!Pv;

PV T= j-=?>VT6=\2>U]7D >VB=OH7IST6D TUPVh^ D8^ V?= ORTQ = I − 2P

; k ]PVB=T6=MHJIgTQ2 = I

TVQT = Q

x ]_~aZ~ QT`V[]2= V eS]2\]!PS^hQT(y ;-Y ]OiVy = (y1, y2) ∈ R2 ;. >UV T=BD OQPgT6= v

j_]7I-=HJIgTh^` TRU]PK@gT RU] ]2=z@]PgPK>UTN@Tz = Qy` ]OiV[PJIShihQT ;

87$7')$28 d = ]" yTV T=hiTOR!TR VT6I-=b = (0 3 0)T `[IK=hiT`E@T= ]!OVT`E@TN@OL=TR VOi]!PS`=T` j_TR V OLRT`

a1 =

2/32/3−1/3

, a2 =

−1/32/32/3

.

` = ]7I-RT=h^j-=]! yTRV OQ]Po@Tb`BI-=hQTjgh^PKTUP-\!TUPK@T=?> jS^&=

a1 TUV a2 ; 87$7')$28 q Y ]OiV

VTUV

W@gT6ITf-` ]2I_`d T` jS^RUT`4RTRV ]7= OQTUh`@T

Rn ; k ]!P!V?= T=EH7ISTdim(V + W ) = dim V + dim W − dim V ∩W.

hQh<IS`VB=T6=RT=?>`BIghV^ Vq^XRTRVTUV

W= T`j4TRV OLRTDTPV»TUP_`TDw]ghQT`@T` D8^ V?= ORT`»VB=OQ^P-\2Igh^OL=T`OQPTb>= OQT6I-=T`TV`BIgj*>6=OiTI-=T` ;

Page 55: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

! #"$&%&'(*)"+,.- /102/436587:9;7=<>9@?BADCD/13E9;9@7FAG3H- IJADK2L89ML89@7(NO/4369K2CD7P<2C=QR<[email protected]?JA=9UKSCD7(<SC=QR<SCD9U7(LV- KS0S9MWX/4A=CDY6?9M?/1C=CDI9RZ,!9U7TNO/4369KSC[7G<SC=QR<SCD9U7TLV- K20S9\W/4A=CDY6?9]?U/4CDC=IU9

A(n× n)7=Q10^AT3E9@7

n7:QR3EKSA=Y6Q1027TLS/40>7

CL89_3H- IU`^K2/4A=Y6Q10a?U/4C[/1?BbADICDY67:A=Yc`^KS9

det(λI −A) = 0d\Ke<fQ1Y60^AGL89;NgKS9MLS9;3H- /436h1iUjSCD9k3EY60SIU/1YECD91l^?J9U36/m7=YEhR0SY$#29M`^KS9;369;0SQO5^/4KL89λI−A

?Q10^A=Y690^A.L89@7(N19@?BAD9KSC[7(0SQR00gKS3c7UlS/4<S<f936IU7nNR9U?BAD9KSC[7T<SCDQ1<SCD9U7n/17D7=Qg?YEI@7noλZ2d\Q102?1lS7:Y

x 6= 09U7:A;KS0pNR9U?JA=9K2Ck<SC=QR<SCD9]L89

A/R7=7=Q8?JY6I]oq3c/NO/13E9UKSCk<SCDQ1<SCD9

λlSQR0r/

Ax = λxZ,!9r?U/43c?JKS3kL89@7

n7:QR3EKSA=Y6Q1027qL89r3H- IU`^K2/4A=Y6Q10s?/4C[/1?JA=IUC=Yc7FADY6`^KS9a9U7:AADC=i@7q?Q1t8AD9K8uvLSiU7X`^KS9

n > 29JAX3E9ADwSIQRC=iUWX9eLV- xMj>9U3.WXQ10^ADC=9`^Ky- Q100S9e<f9K8A9U7=<>IUC=9UCm3E9CDIU7=Q1K2LSC=9X<2/1CzL89U7+CD/RL8Y6?U/4K8uL8iU7+`^KS9

n > 4Z#|\0CD9U?[w29C[?[wS9C[/L8QR02?rL89U7eWqIA=wSQ8L89@7XY~ADIC[/OADYENR9U7`^KSY;<f9CDWX9JA:AD90^AaLV- /4<S<2C=Q8?[wS9UCX?9U7XC[/1?YE0S9@7q9JAe0SQR0LS93E9@7?U/43c?JKS369Cz9Ju8<S36Y6?Y~AD9WX90^A?U/4C@l*or3c/L8YEfIUC=9U02?J9aLS9U7+WXIA=wSQ8L89@7zL89eCDIU7=Q136K8ADYEQR0LS9a7:587:A=iUWq9@7+36YE02IU/4Y6CD9U7mNgKS9@7L2/4027369;?[w2/4<2Y~ADC=9k8lO3c/m?JQR0^NR9CDh19U02?J9T7:9UCD/\Y6?Y2/17=5gWX<8A=Q1A=Yc`RK291Z(0X/4YEAUl13E9@7WqIA=wSQ8L89@7(`^KSYS7=9CDQ10^A#<2C=I@7:9U0RADI9@7<fQ1K2CG369\?U/43c?JKS3L89U7GNO/4369KSC[7(<2C=QR<SC=9@7(7=Q10^AGK8ADYE36Yc7:IU9U7G<>QRKSCG9ugA=C[/4Y6C=9k369U7GCD/R?JY60S9U7GLV- KS0a<fQ1365g0S1WX9;9U0a<2/R7=7D/40^A<>/4Ck3c/W/4A=CDY6?9_?Q1WX<2/1h10S96%

n−1∑

i=0

aiti + tn = 0

9@7FA;369_<>QR3E5g0SRWX9_?U/4C[/1?JA=ICDYc7FADY6`^KS9_L89_36/XW/4A=CDY6?9_?Q1WX<2/1h10S9

A =

0 · · · −a0

1

Z Z Z−a1

0 0

ZZZ0 1 −an−1

D *n #T#$&RF+&+B!!k'.! #zJ QR027:YcL8IUC=QR027T<2/4Ck9ug9UWX<S3E9mK20r7=587FADiWX9mLV- IU`^K2/OADYEQR027kL8YEfIUC=9U0^A=Y69363E9@7nQ1C[L8YE0>/4Y6C=9@7%*CDQ1KSNR9Cn3E9@7TQ102?JA=Y6Q1027

v(t)9A

w(t)l><>QRKSC

t ∈ R+ l8A=9U3E369U7k`^KS9&%dv

dt= 4v − 5w

dw

dt= 2v − 3w

R

Page 56: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

/N19@??JQRWXWq9?JQR02L8YEA=Y6Q1027pYE02Y~ADY6/13E9@7

v(0) = 8, w(0) = 5Z 97=587FADiWX9<f9KSA A=CD9WXY67r7=Q1K27a36/ Q1CDWX9N19@?BADQ1CDYE9U3E369

du

dt= Au ! Z#"%$/N19@?6%

u(t) =

(

v(t)w(t)

)

, u(0) =

(

85

)

, A =

(

4 − 52 − 3

)

& /R?[w2/40^AT`^KS9]3c/z7=Q136K8A=Y6Q10aLS9\3 - I@`^K2/OADYEQR0x′ = ax

9U0pL8Y6WX9027=Y6Q10'"\9U7:Ax(t) = x0e

at l8`RK2YLSYENR9CDh19M7=Y a > 09JA7=9p7:AD/4j2YE36Y67=9p/R7:5gWX<8ADQ4A=Yc`^KS9WX9U0RAXo)(UICDQ7:Ya < 0

l#Q10 ?[wS9C[?[wS9C[/L89@77=Q136K8A=Y6Q10>7z<2/1C:ADY6?KS36YEiUC=9@7zL89p36/Q1CDWX9A =

v(t) = veλt

w(t) = weλt

(0CD9WX<S3c/+*U/40^AnL2/4027 ! Z#"%$Bl2Q10&N1Q1YEA;`^KS9

u(t) = eλt

(

vw

)

⇒ du

dt= λeλt

(

vw

)

= Au(t)

LV- Q+,A

(

vw

)

= λ

(

vw

)

9JAλLSQ1YEA\L8Q10>?- JADC=9+K20S9+NO/4369KSC\<SCDQ1<SCD9+L89

A/R7=7=Q8?JY6I9+/4KNR9U?JA=9K2C;<SCDQ1<SCD9 ( v

w

) Zfd]/40>7;369z?/R7k<SCDIU7=90^AUlQ10?U/43c?JKS369m/4Yc7:IUWq9U0^An369U7nIU3EIUWX90^AD7n<SCDQ1<2C=9@7kL89

A%

λ1 = −1⇒(

v1

w1

)

=

(

11

)

⇒ u1(t) =

(

e−t

e−t

)

λ1 = 2⇒(

v2

w2

)

=

(

52

)

⇒ u2(t) =

(

5e2t

2e2t

)

,!/a3EY60SI@/4CDY~ADIzL8K7=587FADiWX9+Y6Wq<23EYc`^KS9`^KS9zA=Q1KSA=9z?Q1Wzj2YE02/1Y67=Q1036Y60SIU/1YECD9+L89U7]7=Q136K8A=Y6Q1027M<2/4C=A=Yc?JK23EY6iCD9U7u1, u29U7:AM7=Q136K8A=Y6Q10rL89m3 - I@`RK>/OA=Y6Q10rL8YEfIUC=9U0^A=Y69363E9RZS,!/X7:QR3EKSA=Y6Q10&h1I02IC[/4369]7U- I@?JCDY~A;L8QR02?0%

u(t) = αu1(t) + βu2(t)

Q+,α = 3

9JAβ = 1

7=Q10^A;L8IJAD9CDWqY60SIU9U7n<2/1Ck3E9@7k?JQ10>L8Y~ADYEQR027nY60SY~ADY6/13E9@7ZX/. 0 k!%&[*_*21zkpn +%

|\0vL8QR0S0S9UCD/rYc?JYn`^KS93c`^KS9U7z<SC=QR<SCDYEIA=IU7LV- YE0^A=IUC/ Aq7=KSC=A=Q1KSAz<SC[/OA=Yc`^KS9a?Q102?9CD02/40^Az369U7zN/13E9UKSC[7+<SCDQ1<SCD9U79JA_369U7\N19U?JA=9UKSCD7\<SCDQ1<SCD9U7\L89?9C=AD/1YE0S9@7MW/4A=CDY6?9U7UZf,!9U7]L8IWXQ10>7FADCD/4A=Y6Q1027\7:QR0RA]Q1WXYc7:9@7l>9JA_Q107=9CDIJIUC=9UCD/o3 - QRKSNgCD/1h19_L8943qZ & ADCD/10Sh65#"87:9 lS<fQ1K2Ck<S3EK>7;L89+L8IAD/4Y63c7Z;;029TW/4A=CDY6?9T?U/4CDC=IU9

AL89kL8Y6Wq9U027=YEQR0

no]?JQg9<?YE9U0^AD7?JQRWq<23E9ug9@7#Q1KqC=IU93c7*<fQR7D7=iUL89

nNO/4369KSC[7#<SCDQ1<SCD9U70SQR00SI@?J9U7D7D/4Y6C=9UWq9U0^A]L8Yc7FADYE0>?BA=9@7mLS/40>7

CZy,.- 9U027:9UWzjS369XL89q?9U7]NO/4369KSC[7\<SC=QR<SCD9U7\9U7:A]369>=8?@AB8C:@L89

AlV9JA369+0SQ1WjSCD9+L89mQ1Yc7kQD,/4<S<2/1CD/:E6AkKS029+NO/4369KSC;<2C=QR<SC=9

λLS/40>7;?J9+7=<>9@?BADC=9+9@7FAM/1<S<f936Iz7=/GFIHKJLB8MN?KJOMNADMLB8PfZ8,y9

C:Q+RTSVU2=8?@AB8C:QWJFl#0SQ4ADIρ(A)

l*9@7FAq3E9p<S36K27zhRCD/102LWXQ8L8KS369pL89@7zNO/4369K2CD7z<SC=QR<SCD9U7+L89AZ,!/7=Q1WXWX9aL89@7NO/4369KSC[7n<SCDQ1<SCD9U7T9@7FA;IUhR/4369_o3c/XBC:QWAD@aL89m3c/qW/OA=CDYc?J90%

n∑

i=1

λi =

n∑

i=1

aii

Page 57: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W V 9A\QR090L8I@L8KSYEA]L8QR02?+`^KS97=Y!K20S9+W/OADC=Yc?J9+CDI9U3E369+<fQR7D7:i@L89+KS029+NO/4369KSC\<SC=QR<SCD9+?JQ1WX<S369Ju89Rl>7=Q10?JQR0 FKShRKSI9@7FAM/1K27D7:YNO/4369KSCk<2C=QR<SC=9RZSdM9mWX UWq9Rl83E9m<2C=Q8L8KSYEAML89@7nN/13E9UKSC[7n<SC=QR<SCD9U7T9U7:AkIhR/13y/4K PB%@DC:F MOU Q U B_LS9

A%

n∏

i=1

λi = det(A)

' B8B8@U B8MOSVU 3k0y- 9U7:A<2/17q<fQR7D7:Y6jS369p9U0 hRI0SIUCD/13TLy- QRj8A=9U0SYECNO/13E9UKSCD7q9AXNR9U?JA=9K2CD7<SCDQ1<SCD9U7XL89&36/7=Q1WXWX9QRK L8K<SC=Q8L8K2Y~AzL89eLS9K8uWX/4A=CDY6?9U7mL8QR0^A_QR0 ?JQ10202/ EcAm3E9e7=<>9@?BADC=9RlyW/4Yc7m?J9C=AD/1YE0>7m?/17_<>/4C=A=Yc?JKS36YE9UCD7_QD,3E9@7NR9U?JA=9K2CD7T<SCDQ1<SCD9U7n7=Q10^Ak3E9@7kWX WX9U7n<f9CDWX9JA=A=90^AML89+?Q102?3EK2C=90% 7:Y λ

9U7:A_NO/13E9UKSCm<SCDQ1<SCD9L89Alλk 9@7FAmNO/4369KSC_<SCDQ1<2C=9XL89 Ak ! 9U0<>/4C=A=Yc?JKS36YE9UCUlV7=Y A

9U7:A+YE0gN19UCD7=Y6jS3E9Rlλ−1 9U7:AkNO/4369KSCk<SCDQ1<2C=9mL89_3 - Y60gN19C[7=9%$

7:Y λ 9U7:A;NO/4369KSCk<SCDQ1<2C=9_L89 Alλ + α

9@7FA;NO/13E9UKSCk<SCDQ1<SCD9]LS9A + αI

Z& Y

λ9@7FA(KS029;NO/4369KSCP<SCDQ1<2C=9kL89

Al4369M7:587:A=iUWq9

(A−λI)x = 0<>Q^7=7=iULS9nLS9U7(7=Q136K8A=Y6Q10>7P0SQ10X0gKS363E9@7(/4<2<>9U3EIU9U7

@AB8@H C:= ? C:SV? C:@=;L89 A/R7=7=Q8?JY6IU7Po

λZg|\0/4<S<f9363E9=%STH = @=8?KQWAD@X? C:SV? C:@/17D7:Q8?JY6Iko_KS0S9kNO/4369K2C(<SCDQ1<SCD9

λ3E9m0SQO5^/4K&L89

A− λIZ & /L8Y6Wq9U027=YEQR0p9@7FAML8QR02?_/4K&WXQ1Y6027nIh^/4369_o"RZ,!Q1C[7=`^KS9n369U7NO/4369KSC[7<SCDQ1<SCD9U77=Q10^A(L8Yc7FADYE02?JA=9@7lO369U7PNR9U?JA=9K2CD7#<SCDQ1<SCD9U77=Q10^AP3EY60SI@/4Y6C=9UWq9U0^AY602L8IU<>9U02LS/10RA[7Z,.- Y6WX<S36Y6?U/OA=Y6Q10&CDIU?JY6<SCDQ8`RK29M9U7:An/4K27D7=91l8Q10&<f9027=9C[//1Kp?U/17kL89_3c/zW/OADC=Yc?J9_YcL890^ADY~ADI]<fQ1K2C;7- 90r?JQR0^NO/1YE02?C=9RZ

& Y]369U7nN19@?BAD9KSC[7<SCDQ1<SCD9U7

xi, 1 ≤ i ≤ n<>9UKSN19U0RA6 JA=CD9?[wSQRY67=Y67e3EY60SIU/1YECD9WX9U0RAeY602L8IU<>9U02LS/10RA[7lGYE3c7QRC=WX9U0RAkKS029mWX/4A=CDY6?9

X = [xi]0SQ10r7:Y60ShRKS3EY6iCD91l89A;Q10r/ %

X−1AX = Λ = diagλ1 · · ·λn.0p9f9AUlAX

9U7:AT3c/zW/4A=CDY6?9\L8QR0RAn369U7n?JQR3EQR0S0S9U7T7=Q10^AG369U7TN19@?BAD9KSC[7Axi = λxi

`^KSY9U7:ATj2YE9U0pIUhR/4369\oXΛ

Z|\0L8YEATL2/4027(?9M?/R7(`^KS9A@=BKMNQ STU QWJOMN=%QJN@m9AG?J9A:A=9\<SCDQ1<SCDYEIA=In0S9;<f9KSAT7U- I@?JCDYECD9k`RK!- /NR9U?n3c/_WX/4A=CDY6?9LS9U7GN19U?JA=9UKSCD7G<SCDQ1<SCD9U7UZ1.3E369]?U/4C[/1?JA=ICDYc7:9k369M/1Y~Ak`^KS9

X/+<[email protected]?BAD9KSC[7(<2C=QR<SC=9@7(9An`^KS9_?J9@7NR9U?JA=9K2CD7T<SCDQ1<SCD9U7n7=Q10^Ak3EY60SI@/4Y6C=9UWq9U0^AnY602L8IU<>9U02LS/10RA[7Z|\j27=9CDNO/OADYEQR0%YE3k9ugYc7:A=9pLS9U7qW/OA=CDYc?J9U7q`^KSYk0S9r7:QR0^A<2/R7qLSY6/1h1Q10>/436Y67D/4jS369U7 ! Q10 3E9@7X/4<S<f9363E9&W/OADC=Yc?J9@7KP @A+B8M @=%$JZS(363E9@7;7D/OA=Yc7:Q10^A;/4K8upLS9K8up?Q102LSY~ADYEQR027k7:K2YENO/40^AD9U7 %

YE3!9Ju8Y67:A=9+LS9U7nNO/4369KSC[7n<SCDQ1<SCD9U7TWKS3~ADYE<23E9@7 36/&L8Y6WX9027=YEQR0L8K7:QRK27:b 9@7:<2/R?J9+<2C=QR<SC=9/17D7:Q8?JY6Iz9@7FAm7:A=CDY6?JA=9WX9U0RA_Y608ICDYE9UKSCD9oe3c/pWzKS3EA=Y6<S36Y6?Y~ADIXLS93c/NO/4369KSCk<SCDQ1<2C=9RZxTA=A=9U0RADYEQR0ylV369q7=9KS3#/1Y~A+L89q3H- 9Ju8Yc7FAD902?9qLS9qNO/13E9UKSCD7\<SCDQ1<SCD9U7\WzK23~ADYE<S369U7]0S97=K<XAm<2/17]oaY6WX<S36Y6`^KS9UC_`^KS9X3c/W/4A=CDY6?9.7:QRY~AL8IJ9@?BADYENR9 ![q 3c/;W/4A=CDY6?9(Y6LS90^A=YEA=IT<2/4C*9Ju89WX<S369%$JZ%;M0z9ug9UWX<S3E9.AF5g<SY6`^KS9TL89.W/4A=CDY6?9.L8IJ9@?BADYENR99@7FAA =

(

0 10 0

) l`^KSYn<>Q^7=7=iUL89a36/N/13E9UKSCq<SC=QR<SCD9aL8Q1K2jS3E97SZ,y9@7zNR9U?BAD9KSC[7+<2C=QR<SC=9@7z7=Q10^AqLS9a3c/Q1CDWX9

x =

(

α0

) l8Y6367;90ShR902L8CD90^A;L8QR02?]KS07:QRK27Fb 9U7=<2/R?J9\LS9+L8YEWX9U027:Y6Q10I"1ZA0y- 9@7FA;L8QR02?_<2/R7kL8Y6/1h1QR02/436Y67D/4j23E9RZ

d]/4027p?J9UC:A[/4Y6027e?/R7l.YE3\9U7:AeC=9U36/4A=Y6N19WX9U0RA/R?JY63E9Ly- Y608ICD9Cp7:KSCp36/NO/13E9UKSCeQ1K3c/ 02/4A=KSCD9L89U7eN/13E9UKSC[7<2C=QR<SC=9@7nLV- K20S9mW/OA=CDYc?J9A%

7:Y A 9U7:AkLSY6/1h1Q10>/43691lg3E9@7kNO/4369KSC[7T<SCDQ1<SCD9U7n7=Q10^Ak3E9@7kI36IWX90^A[7kL8Y6/1h1QR02/4K8u 7:Y A 9U7:ATADC=Yc/402h1KS3c/4Y6C=9Rlg3E9@7kNO/4369KSC[7T<SC=QR<SCD9U7n7:QR0^AnIUhR/13E9UWq9U0^An369U7nIU3EIUWX90^AD7;LSY6/1h1Q10>/4K8u 7:Y A 9U7:An02Q107:Y60ShRKS3EY6iCD91lg369U7kNO/13E9UKSCD7n<2C=QR<SC=9@7n7:QR0RAnADQ1K8AD9U7kL8YEfIUC=9U0^A=9U7kLS9 7 7:Y A

9U7:A\Q1C=A=w2Q1h1QR02/43691l23E9@7MNO/4369K2CD7M<SC=QR<SCD9U7;QR0^A\<fQ1KSC]WXQgLSKS3E96"z9A]Y63*9@7FA]<>Q^7=7=Y6jS3E9qL89?[w2Q1Yc7:Y6CMKS029j2/17=9mL89mN19@?BAD9KSC[7n<SCDQ1<SCD9U7TQ1C=A=w2Q10SQRC=WXIU7 7:Y A 9U7:A#7=5^WXIA=CDY6`^KS9RlU7=9U7*N/13E9UKSC[7y<SCDQ1<SCD9U7!7:QR0RA#C=IU9363E9@7Z@,y9@7!NR9U?BAD9KSC[7y<2C=QR<SC=9@7y/17D7=Qg?YEI@7!o;LS9U7*N/13E9UKSC[7<SCDQ1<SCD9U7kL8Yc7:A=Y602?BAD9U7k7=Q10^A;/436Q1C[7GQRC:ADwSQ1hRQ102/1K8uVZ,!9]LS9CD0SYE9UCG<fQ1Y60^ATY6WX<S36Y6`^KS9]90p<2/4C=A=Yc?JKS36Y69Cn`RK29MADQ1K8AD9\W/OADC=Yc?J9]CDI936369\7=5^WXIA=CDY6`^KS9\9U7:AnL8Yc/4hRQ102/13EYc7D/4jS369;<2/1CK20S9mW/OA=CDYc?J9mQ1C=A=w2Q1h1QR02/43696%

Λ = QT AQl QD,

Q9@7FAkQRC=WXIU9m<2/4C

nN19U?JA=9UKSCD7n<2C=QR<SC=9@7nQ1C=A=wSQR0SQ1CDWXIU7UZ2|\0r//13EQRCD7T3c/ Q AB8SVC:MN=%QB8MOSVU =8?@AB8C:QWJN@eLy- KS029mWX/4A=CDY6?9m7:5gWXIJADC=Yc`^KS9_C=IU9363E90%

A = QΛQT

Page 58: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

(0rCDIU7=KSWXI_LS9+?J9m`^KSYy<SCDIU?iUL890% J#MNU @C:=%M MNJNMNB8P@=BGJNMOP@Q H QWJN@H C:= ?KC:S ?KC:@= JNQ MOQ SVUKQWJNMO=8Q KMOJNMLB%P6@=8B>JOMNP@ QWH @A+B8@H C = ? C:SV? C:@=

X ) )JJ)[#%p$&k,(- Q1j F9U?BADY~n9U7:Am902?Q1CD9XKS0S9XQ1Yc7mL89XADCD/1027FQRC=WX9UC]K20S9W/OA=CDYc?J9X<2/4C+LS9U7]A=C[/40>7FQRC=W/OADYEQR027_7:Y6WX<S369U7m9U0KS0S9kW/4A=CDY6?9nLSQ10^APQ10X?JQ10202/ EcAP3E9@7PNO/4369KSC[7#<SC=QR<SCD9U7Ul?1- 9@7FA=b o4b LSYECD91lKS0S9kW/4A=CDY6?9GADC=Yc/402h1KS3c/4Y6C=9GQ1KXL8Yc/4h1QR02/43691Z,y9@7nA=C[/40>7FQRC=W/OADYEQR027T`RK2YyW/4Y60RADYE9U0S0S9U0RAk369+7:<f9U?JA=CD9mLV- K20S9mW/OA=CDYc?J9m7:QR0^AkLS9U7;7=YEWXY63EYEA=K>L89U7UZ

!#"%$& '$&()*+,P.-UKMLB8MOSVU0/)13246587:9<;>=?3@BA3CD5FEGCF=:@B@FHI5FE

A58?

BEFJ8KL?NMOAP?Q5DESROT:UWVYX[ZVYXTRGE]\ A_^L5B9`AaEI?Q57OKY5G;b=`?[@cA[CF5

SKYJ8K

EcA_K`dY7^aAfeF@F5g?Q5F^_^5gh`7]5B = S−1AS

,!/A=C[/40>7FQRC=W/OADYEQR0 L89ANR9C[7

B9U7:AqK20S9 =8MOF MNJOMLB8H K@>ZV.0 3 - I@?JCDYENO/10RAX7:QRK27q36/Q1CDWX9

AS = SBlQ10CD9JA=CDQ1K2N19KS029hRI0SIUCD/13EYc7=/4A=Y6Q10 L8936/ L8I/#202Y~ADYEQR0 L89@7aNO/4369K2CD7a<SC=QR<SCD9U7e9ArLS9U7pN19U?JA=9UKSCD7a<SCDQ1<SCD9U7UZn|\0 /LV- /4Y63E369K2CD7n369mC=I@7:K23~A[/OAnQ102L2/4WX90^AD/13 %

i C:S ? SV=%MLB%MNSTUj/)13246587:9k;>=?3@BA3CD5FElEF5m;kn:^o=(n:^5FEpJ8KL?q^f5DEl;srm;s5FEutD=^f5:7@BE'v@FJvL@F5DEwyxz|~L]~|g7=Q1YEA

xKS0XN19@?BAD9KSC(<2C=QR<SC=9nL89

A/R7=7=Q8?JY6Iko_36/_NO/4369K2CP<SC=QR<SCD9

λZg|\0/_L8Q10>?

Ax = λxl`^KSY!7U- I@?JCDY~A

SBS−1x = λxl2?9+`^KSYyN19UK8A;L8Y6C=9m`^KS9

λ9@7FAkNO/4369KSCk<2C=QR<SC=9_L89

B/R7=7=Q8?JY6I]/1KpNR9U?BAD9KSCk<2C=QR<SC=9

S−1xZ

,(- Y60RADIC JA;L89+?9U7TA=C[/40>7FQRC=W/OADYEQR027T9U7:A;L8Q1KSj23E9&% 3E9@7nN/13E9UKSC[7n<SC=QR<SCD9U7n7:QR0^AnY602?[w2/10Sh1IU9U7 907:K2<S<>Q^7=/10^An369U7nNR9U?JA=9K2CD7n<SCDQ1<2C=9@7G36Y60SIU/1YECD9WX90^AkY602L8I<f90>LS/40^AD7UlS3c/X7:Y6WXYE36Y~ADK2L89+/R7=7=Q8?JY6I9moq36/qW/ObA=CDY6?9

XL8Q10^Am369U7m?JQR3EQR0S0S9@7]7=Q10^Am369U7_N19@?BAD9KSC[7]<SC=QR<SCD9U7\A=C[/4027:Q1CDWX9

A9U0KS029WX/4A=CDY6?9qLSY6/1h1Q10>/4369L8Q10^Ak3E9@7kI36IWX9U0RA[7kL8Yc/4h1QR02/4KSua7:QR0RAk369U7nNO/4369K2CD7n<SCDQ1<2C=9@7TLS9

A%X−1AX = Λ

ZQR0RADC=QR027pWX/1YE0^AD902/10RAr7:K2C&K20 9ug9UWX<S3E9`^KS9L89K8u W/4A=CDY6?9U7r7:9UWzjS3c/4j23E9@7aCD9<SCDIU7=90^AD90^Ap3c/vWX UWq9A=C[/40>7FQRC=W/OADYEQR0p36Y60SIU/1YECD9m7:KSC\L89K8urj2/R7:9@7kL8Y~ICD90^AD9U7:%27=Q1YEA

P36/XW/4A=CDY6?9_LS9+<SCDQ! F9U?JA=Y6Q10rLS/40>7

R2 7:K2C;36/L8CDQ1YEA=9LLV- /40ShR3E9

θ%

P =

[

cos2(θ) cos(θ) sin(θ)cos(θ) sin(θ) sin2(θ)

]

= uuT /N19U?u =

(

cos(θ)sin(θ)

)

& YPQR0/4YEAmWX/1YE0^AD902/10RA\A=QRKSC=029C]36/pj2/17=9X?/402Q10SYc`^KS9zQRC:ADwSQ10SQRC=WXIU9zLV- KS0/10Sh1369θlf3c/p<SC=Q" F9U?JA=Y6Q10L89UNgYE9U0RAW/4Y60^A=90>/40^AkKS0S9m<SCDQ! F9@?BA=Y6Q10r7=KSCk3H- /Ou89_wSQRC=Y#(QR0RA[/439JA;7U- I@?JCDY~A

Q =

[

1 00 0

]

Q1KSCk<>/17D7:9UC;L89Po

Ql8Q10&K8ADYE36Y67=9m3c/XWX/4A=CDY6?9mL89_C=Q1AD/OADYEQR0&Ly- /10Sh1369

θ

R =

[

cos(θ) − sin(θ)sin(θ) cos(θ)

]

,y9+?[w>/40ShR9WX90^AML89mj2/R7:9m7=9_A=C[/1L8K2Y~A;<2/1C %27=Yx9@7FA;369mN19@?BA=9UKSC;L89z?QgQ1C[L8Q10S02I9U7nLS/1027k3c/qj2/17=9+L89+L8IU<2/4C=AUl9JA

u369mN19@?BAD9KSC;L89m?Q^QRCDLSQ10S0SIU9U7nLS/1027n36/Xj>/17=9]LV- /4CDCDYENRI91lgQR0&/

x = Ru

Page 59: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

6) 4 & Q1YEA

y3c/z<SCDQ! F9@?BADYEQR0eL89

x7=KSC

LLS/40>7.3c/+j2/R7:9]L89_L8I<2/1C:A@lR9A

v7=9U7n?JQgQRCDL8QR0S0SIU9U7(LS/1027T36/zj2/17=9\Ly- /1C=CDYENRI9RZxM3EQRCD7

y = Px

Rv = PRu

⇒ v = R−1PRu = Qu

9AML8Q10>?Q = R−1PR

9A;L8Q102?P9JA

QQR0RAk369U7nWX WX9@7nN/13E9UKSC[7G<2C=QR<SC=9@7Z Q1WXWX9

Q9@7FA;L8Yc/4hRQ102/13E9RlgQ10aCD9JADC=QRKSN19]Yc?JYV369]CDIU7=KS3EAD/4A?Q10S0gK %SADQ1K8AD9mWX/4A=CDY6?9mL89+<SCDQ! F9@?BA=Y6Q10&7:K2C;KS0S9+L8CDQ1YEA=9+L2/4027

R2 /q<fQ1KSCMN/13E9UKSC[7n<SC=QR<SCD9U7 "_9JA72l89JAM<fQ1K2CNR9U?JA=9K2CD7k<2C=QR<SC=9@7;/17D7=Qg?YEI@7k369U7M?Q136Q10S029U7;L89z36/W/4A=CDY6?9zL89+CDQ4A[/OA=Y6Q10

Rl>?1- 9@7FA=b o4b LSYECD9m3E9@7ML8Y6CD9U?BADYEQR027;L89

L9AL⊥ C=9@7:<f9U?JA=Y6N19UWq9U0^AUZ

lp ) '& " p"")&(+ S '&(;M0S9/13~AD9CD02/OADYENR9<SC[/OADY6`^KS9/4K8u /436h1QRC=YEA=wSWX9@7aL89?/136?KS3\/1<S<SCDQg?[w2IL89U7pNO/4369KSC[7p<SC=QR<SCD9U7a`^KS90SQ1K>7LSIU?JCDY6C=QR027+<S36K27q36Q1Y60v9U7:AX3E9pA=wSIUQ1CDiWX9p7=KSY6NO/40^AX`^KSYn<f9CDWX9JAL89p36Q8?/13EYc7:9UCz369U7qNO/4369KSC[7<SC=QR<SCD9U7zL2/4027qL89U7LSY67D`^KS9U7UlSL8YEAD7;L8Yc7D`RK29U7kL8943]9UCD7=wSh1QRC=Y60yl8L8K&<S3c/40?JQRWq<23E9ug9RZ

PSVCF @ /)132 AGJ:K @D5vL@DHFEF5mKL?Q5b7KY5g;b=`?[@cA[CF5A J`7 ?J*7?5g;>=?3@BA3CD5 EF5m;kn:^f=n:^5

A EFJ*7.E^f=[J8@B;s5A = diagd1 · · · dn+F J F

5DE?'7OKY5;b=`?[@cA[CF5M*5MOAf=8dJ8K =^f5!KL7^a^f5 =^fJ:@BE ^5EQvN58CI?[@F5M`5 A5DE? CDJ8KL?Q5FKL7

M*=:K(E^F\ 7OKAfJ:K M*5FEMOAoEh*7*5FEDi, 1 ≤ i ≤ n

M(76vL^f=:K CFJ:; vL^f5m9 5 ?Q5F^aE6h*7*5

Di =

z ∈ C, |z − di| ≤n∑

j=1

|fij |

wyxz|~L*~L| \7=Q1YEAλKS029NO/4369K2C&<2C=QR<SC=97:K2<S<>Q^7:IU9LSY~ICD90^A=9L89@7

di, 1 ≤ i ≤ nZkxM3EQRCD7

(D − λI) + F9U7:A;7:Y60Sh1K23EY6iCD9\9A;LV- /1<SCDiU7n369\ADwSIQRC=iUWX92Z "Rl8Q10p/q3c/qWX/! FQ1C[/OADYEQR0 ‖(D − λI)−1 + F‖ ≥ 1

Z,!9m?[wSQ1YEupL89m3c/q0SQ1CDWX9 ‖.‖∞ QRKSC=02Y~A;369_C=I@7:KS3EAD/4A;?[wS9C[?[wSI1Z;M0S9/4<S<S36Yc?/OADYEQR0Y60^A=ICD9U7D7D/40^A=9qL89?J9qC=I@7:KS3EAD/4A]9@7FAm3 - 9@7FADYEW/4A=Y6Q10LS9U7]NO/4369KSC[7]<SC=QR<SCD9U7]LV- K20S9XWX/4A=CDY6?9QRj8A=9U0gKS9_90r<>9UC:ADKSCDj2/40^AkKS0S9mW/OADC=Yc?J9mL8QR0^AnQR0&?Q10S0>/ EcAn369+7=<>9@?BA=CD91Z

@F ?JN@ /1[2

A =

1 0.1 −0.10 2 0.4−0.2 0 3

M`J8KL?q^f5DE6tD=:^587@mE vL@FJvL@D5FEEmJ:KL?qEcAP?7]HI5FEuM]=K(E^5FEMOAoEDh`7]5DElE7OA[tD=:KL?PE "! A dY7@D5 "# D1 = z ∈ C, |z − 1| ≤ 0.2D2 = z ∈ C, |z − 2| ≤ 0.4D3 = z ∈ C, |z − 3| ≤ 0.2

Page 60: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

7

Z " L8Yc7=`^KS9@7kL8943]9C[7=wSh1QRC=Y60

X[ +J'T%&$pk]+Jk%p # & #"+p k $:%p& _! )J'G yb T! )X%& T&"$p $p q+'T"&)

,(- Y60RADIC JA.<2C=Y602?JY6<2/132L89U7.W/OA=CDYc?J9U7.C=IU9363E9@7(7=5^WXIA=CDY6`^KS9@7P9U7:AG`^Ky- YE3>9Ju8Y67:A=9MKS0S9Mj>/17=9;L89;N19@?BAD9KSC[7P<SCDQ1<SCD9U7Q1C=A=w2Q10SQRC=WXIU7UZ.|\0<>9UK8AaL8Q10>?r36/L8Yc/4hRQ102/13EYc7:9UC<>/4CKS0S9rADCD/1027:Q1CDWX/4A=Y6Q10 Q1C=A=wSQRh1Q10>/4369%.7=Q1Y690^AAKS0S9A=9U3E369W/OADC=Yc?J9q9JA

Q36/&W/OADC=Yc?J9XQ1C=A=wSQRh1QR02/4369zL8QR0RAm369U7m?Q136Q10S0S9@7_7:QR0RA_369U7_N19@?BAD9KSC[7]<SC=QR<SCD9U7_L89

Aly/13EQRCD7

QT AQ = diagλ1 · · ·λnZ

,!/XWXIA=wSQ8L89qL89^/1?JQRjSYy9@7FAMK20S9+WXIJADwSQ8L89zLV- I36YEWXY602/OADYEQR07=5^WXIA=CDY6`^KS9mYEA=IUCD/4A=Y6N19mK8ADYE36Y67D/40^A\LS9U7M7=Y6WqY636Y~bA=K>L89U7kQRC:ADwSQ1hRQ102/13E9@7% 36/qW/OADC=Yc?J9]A=C[/40>7FQRC=WXI9\A=9U02L&N19UCD7nKS029]W/4A=CDY6?9_LSY6/1h1Q10>/4369 3E9m<SCDQ8L8KSYEA;L89U7nADCD/1027:Q1CDWX/4A=Y6Q1027GQ1C=A=w2Q1h1QR02/4369U7.A=9U02L&N19UCD7T3c/XWX/4A=CDY6?9mL89U7nNR9U?BAD9KSC[7T<SCDQ1<SCD9U7UZ

[ l &(c$6" c N` S#+ `&( "

9T<SC=Y602?YE<f91lj2/R7:IT7:KSCL89@7*CDQ4A[/OADYEQR027!7=K2??9U7D7:Y6N19@7l@7:9UCD/MYE3636K27FADC=I.A=QRK8ALy- /1j>QRCDL+7=KSC#KS0S9TW/OA=CDYc?J9 ! × D$ %7=Q1YEAA36/X7=Q1K27:b W/4A=CDY6?9 ! × +$G7:5gWXIJADC=Yc`^KS9

A =

(

app apq

apq aqq

)

apq 6= 0.

& QRY~AR3c/XW/OADC=Yc?J9mL89_CDQ4AD/4A=Y6Q10rLV- /40ShR3E9 −θ

%

R =

(

cos θ sin θ− sin θ cos θ

)

.

Q1KSCkIU3EY6WXYE029C;3 - IU3EIUWX90^Aapq

0SQR0&LSY6/1h1Q10>/43 l^QR0L8IJAD9CDWqY60S9θA=9U3!`^KS9

RT AR =

(

∗ 00 ∗

)

,y9+?U/43c?JKS3yLSQ10S0S9cotg(2θ) =

aqq − app

2apq

Page 61: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

K 4 K 4k\ ) K' 4 4 + " QR027=Y6L8IUC=QR027kW/4Y60^A=9U02/40^AMKS029zW/OADC=Yc?J9z7=5gWXIJA=CDYc`RK29

n × n AA=9U3E369`^KS93H- I36IWX90^A

apq7:QRY~AM02Q100gKS3 Zf,*/ADCD/1027:Q1CDWX/4A=Y6Q10pQ1C=A=wSQRh1Q10>/4369

ΩAD9363E9+`^KS9

A′ = ΩT AΩl2/NR9U?

a′pq = a′

qp = 09U7:A

Ω =

1 Z Z Zcos θ sin θ− sin θ cos θZ Z Z

1

pq

|\0qN1IUC=Y29n`^KS9k7:9UKS3E9@7#3E9@73EY6h10S9@7#9JA.?JQR3EQR0S0S9@7p9JA

qL89

A7=Q10^APWXQ8L8Y>I9U7UZ(0X<>Q^7=/10^A

c = cos θls = sin θ

lt = tan θ

lS3c/XWXY67=9moFQRKSC;7U- I@?JCDY~Aa′

pj = capj − saqj , j 6= p, q

a′qj = caqj + sapj , j 6= p, q

a′pp = app − tapq

a′qq = aqq + tapq

;M0e?[wSQ1YEu?J3c/17D7:Yc`^KS9k<>QRKSCp9A

q9U7:AG?J9U3EK2Y>`^KSY2<f9CDWX9JAGLy- IU3EY6WXYE0S9UCG3 - I36IWX90^A.0SQR0eL8Yc/4h1QR02/43SLS9;<S36K27.hRCD/102LWXQ8L8KS369 |apq | = max

i6=j|aij |

l[ SS")& '"L"& QRY~A Ak

3c/s7:KSYEA=9L89WX/4A=CDY6?9U7p90ShR902LSC=IU9<2/1Cp3 - /13EhRQ1CDY~ADwSWX990 I36YEWXY602/40^Aov?[w2/R`^KS9YEA=IUCD/4A=Y6Q10K20 I36IWX90^A&0SQ10 L8Yc/4hRQ102/13M0SQR0 0gKS3 ! A0 = A$a9JAr7:QR0 7:5gWXIJADC=Yc`^KS91Z Q1WXWX93c/A=C[/4027:Q1CDW/OADYEQR0s9@7FAQRC:ADwSQ1hRQ102/13E9RlR3c/q0SQ1CDWX9mL89SCDQ1jf90SY6K27kL89m3c/qW/OA=CDYc?J9_9U7:AM?JQR027:9UC=NRI9 ![q ?[w>/4<SYEA=CD9$

i,j

a′2ij =

i,j

a2ij

/4Yc7k?J9_WX WX9mC=I@7:K23~A[/OAk9U7:A;NgCD/1Yf<fQ1K2C;36/qW/OADC=Yc?J9 ! × +$.ADCD/1027FQRC=WXIU9]?Y~b L89U7D7:K>7Z8d\Q102?a′2

pp + a′2qq = a2

pp + a2qq + 2a2

pq QRWXWq9e7=9K23E9@7_3E9@7m3EY6h1029U7p9JA

q7=Q10^AmWqQ8L8Y2IU9U7Ul!?J9@7mC=I@7:K23~A[/OAD7_Y6WX<S36Y6`^KS9U0RAz`^KS93c/7=Q1WXWX9XLS9U7+?U/4CDC=I@7LS9U7_I36IWX9U0RA[7mL8Yc/4h1QR02/4KSu/4KShRWq9U0^A=97FADC=Yc?BAD9WX90^A+L89X3c/pNO/13E9UKSC

2a2pq

ly9Az`RK29q3c/r7:QRWXWq9L89@7m?/1C=CDIU7]L89U7IU3EIUWX90^AD7n0SQR0L8Y6/1h1QR02/4K8uaLSYEWXY60^K29_7:A=CDYc?BA=9UWX90^AML89m3c/WX UWq9+`^K2/10^A=YEA=I1Z|\0rL8IUWqQR0^A=CD9_/13EQRCD7T369_A=wSIUQ1CDiWX9m7:K2YENO/40^A PSVCF @ /)1 =6EI7AP?Q5uM*5G;b=`?[@cA[CF5DE Ak

5mKd`5mK)MO@FHI5 v)=@'^o=6;bH:?! YJM*5uM*5 =*CDJ`n:ACc^f=EBEA[h`7]5CFJ:KYt:5m@cd`5t:5m@mE7KY5;b=`?[@cA[CF5 MOAf=8dJ8K =^f56CFJ:KY?5mK =KL?G?J*7?5FE^5FEutD=:^587O@BE'vL@DJDv@F5DEM`5

AE7@l^f=sMOAf=8d`J:K =:^5

wyxz|D~L*~| 2NRQ1Y6Cn<2/4Ck9ug9UWX<S3E9 Yc/4CD3E9A-5 :9 Z|\0 <>9UK8AIhR/13E9UWX90^AqL8IUWXQ10^A=CD9Cq`RK291lP7:Yn369U7NO/4369KSC[7z<2C=QR<SC=9@77=Q10^AA=Q1KSA=9U7XL8Yc7:A=Y602?BAD9U7Ul#3c/7=KSYEA=9rL89U7W/4A=CDY6?9U7Qk = Ω1 · · ·Ωk

?Q10gN19UC=hR9MNR9C[7G3c/XW/OADC=Yc?J9_Q1C=A=w2Q1h1QR02/4369\?Q10^A=9U02/40^Ak369U7nN19@?BAD9KSC[7T<SC=QR<SCD9U7UZ" BB%@U B8MNSTU #SK20&IU3EIUWX90^AM/40S0gKS36I_<>9UK8A;CD9UL89UN1902YECn0SQR0&0gKS3y/1K8uaYEA=IUCD/4A=Y6Q1027k7=KSY6N/10^A=9U7UZ

l[ %$ &('+]")&) N` S+

,y9PADC=Y^L8K+<S36K27!h1C[/402L]I36IWX9U0RA*<2/4CDWXY n(n−1)/2?Q1t8A[/40^A!CD93c/OA=Y6N19UWX90^A!?[wS9C@l Q10]3EKSY^<SCDIJiUC=9(LV- /4KSA=CD9U77FADCD/4A=IUh1Y69U7n<S36K27nIU?Q10SQRWXY6`^KS9@7 ! j2/136/5^/4h19]?J58?J36Yc`RK29]QRK&?[w2Q1YEup/N19@?\7=9KSY63 $

Page 62: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

,!/]WXIJA=w2QgLS9;L89 R/R?JQRjSYg<SC=I@7:9U0^A=9kLV- 9JuS?J9U3E3690^A=9@7<>9UC:QRC=W/102?J9@7*<fQ1K2C(L89U7PW/4A=CDY6?9U7#<S369Y60S9U7(LS9n/4Y6jS3E9L8YEWX9U027:Y6Q10 ! AF5g<SYc`^KS9WX90^AmY608ICDYE9UKSCD9Xo "87D7D$JZfd]/4027_369?/R7\hRI0SIUCD/13HlfQ103EK2YP<SC=IICD9C[/e3c/aWXIA=wSQ8L89 !q ?JYEb /4<SCDiU7 $BZX +J'T%&z$& 'Gn **+)J& +k'(*k%p v& #"+p k MJk r%p)!!zp'Tkv)[#n #kk l "! " "

Q1KSCTKS0S9]W/OADC=Yc?J9]7:5gWXIJADC=Yc`^KS9Alg369]`^KSQ1A=Y690^ATLS9 /5g369Y6h1w9@7FAn3E9\CD/1<S<fQ1C=ATL8I20SY<>QRKSCGA=Q1KSATNR9U?BAD9KSC

x 6= 0<>/4C

ρA(x) =xT Ax

xT x|\0 N1IUC=Y29Y6WXWqI@L8Yc/OA=9UWX90^Aq`^KS9a7=Yx9U7:AzN19@?BAD9KSC+<2C=QR<SC=9Rly369a`^KSQ4ADYE9U0RAqL89 /5g369Y6h1wQ1KSCD0SYEA3c/rNO/4369KSC<SCDQ1<SCD9m/17D7:Q8?YEIU9S9U0p9f9A

Ax = λx⇒ xT Ax = λxT xZ

& Yλ19JA

λn7:QR0RAGC=9@7:<f9U?JA=Y6N19WX9U0RAT36/+<S3EK>7.<f9JADY~AD9\9AG3c/+<S36K27Gh1C[/402L89MN/13E9UKSC.<2C=QR<SC=9\L89

AlR9A

x1 l xn 369U7N19@?BAD9KSC[7n<SCDQ1<SCD9U7n/17D7=Qg?YEI@7lgQR0&/qIh^/4369WX90^Ak369U7nCDIU7=KS3~A[/OA[7k7:KSY6NO/40^AD7 λ1 = ρA(x1) = min

x∈RnρA(x)

λn = ρA(xn) = maxx∈RnρA(x)

d\9m<S3EK>7l27=Yy369U7nNO/4369K2CD7n<SCDQ1<2C=9@7T7=Q10^AkC[/40ShRI9U7TLS/1027k3 - QRCDL8CD9_?JCDQ1Yc7=7D/40^AUlgQR0&/λ1 = min

Si

maxx∈Si

ρA(x)

λn = maxSi−1

minx∈S⊥

i−1

ρA(x)

Q+,Si9@7FAkKS07=Q1K27:b 9U7=<2/1?9]`^KS9U36?Q102`^KS9mL89mL8Y6WX9027=YEQR0

iZ

,y9X7=Q1K27:b 9@7:<>/1?J9Si<fQ1K2C\369U`^KS9U3#369X`RK2Q4A=Y690^AmL89 /5g3E9UYEhRw9U7:A]W/Ou8Y6WzKSW9@7FA]3E9X7=Q1K27:b 9@7:<>/1?J9z<SC=QR<SCD9/17D7=Qg?YEIG/4K8u

i<SC=9UWXYEiUC=9@7VNO/4369KSC[7y<SCDQ1<2C=9@7Z,!9G7=Q1K27:b 9U7=<2/1?9P<>QRKSC!369U`^KS9U3RY63R9@7FA*WqY60SY6WzKSW 9@7FA#Q1C=A=wSQRh1QR02/434/1K7=Q1K27:b 9@7:<>/1?J9M<SC=QR<SCD9\/17D7=Qg?YEI\/4K8u

i− 1<SCD9WXY6iCD9U7GN/13E9UKSC[7.<SCDQ1<2C=9@7Z - 9@7FAnL8Q10>?M369]7=Q1K>7Fb 9U7=<2/1?9;902h190>L8C=I<2/1Ck3E9@7

n− i + 1NR9U?JA=9K2CD7n<SCDQ1<2C=9@7T/R7=7=Q8?JY6IU7no λi · · ·λn

Z l p p" "+ $ +*+ L"+%`)&"+

,!/+WqIA=wSQ8L89]L89U7G<SKSYc7D7=/102?J9@7PY~ADICDI9@7(<f9CDWq9ATL89]?/136?KS3E9UC(369MN19@?BAD9KSCG<SC=QR<SCD9;/17D7:Q8?JY6IMom3c/m<S36K27Gh1C[/402LS9NO/4369KSCk<2C=QR<SC=9RZ& KS<2<>Q^7:QR027

A7:5gWXIJADC=Yc`^KS9mL89mNO/4369KSC[7n<SCDQ1<SCD9U7TQRCDL8QR0S0SIU9U7n7:9U3EQR0

|λ1| ≤ · · · |λn−1| < |λn||\0p?Q1027=YcL8iCD9M3 - YEA=IUCD/4A=Y6Q10&7:KSY6NO/40^A=9]L8I >0SYE9moz<2/4C=A=Y6CkLV- K20aN19@?BAD9KSCnY60SY~ADY6/13q0L8Q10S02I1l^A=9U3y`RK29 ‖q0‖ = 1

lg9JAq00!- 9@7FA;<2/R7nQ1C=A=wSQRh1QR02/43o

vn l23E9_N19@?BAD9KSCk<SCDQ1<2C=9_/17D7:Q8?YEImoq36/q<S36K27khRCD/102L89]NO/4369KSCk<SCDQ1<2C=9]Y67=Q136I9 λn

xk+1 = Aqk

qk+1 =xk+1

‖xk+1‖P/4CkCDIU?KSC=CD90>?J91lgQR0pWXQ10^ADC=9m`^KS9

qk =Akq0

‖Akq0‖

Page 63: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W K 4 9AM?JQRWqWX9_369U7nN19@?BAD9KSC[7n<SCDQ1<SCD9U7 v1 · · · vn Q1CDWX90^AkKS0S9mj2/R7:9mL89 Rn l8QR0&<f9KSA;IU?JCDY6C=9

q0 =

n∑

i=1

αivi, αn 6= 0

9AAkq0 = αnλk

n

(

vn +n−1∑

i=1

αi

αn

(

λi

λn

)k

vi

)

,!Q1C[7=`^KS9k → ∞ l!369U7mC[/4<2<>QRC:A[7 ( λi

λn

)k A=90>L890^AzNR9C[7 7r<>QRKSCi 6= n

l*?9a`^KSY.7=YEhR0SY29a`^KS93c/7:K2Y~AD9eL89U7YEA=IUC=I@7 qk

?JQ10gNR9CDh19mN19UCD7\369zN19@?BAD9KSC]<SCDQ1<SCD9vn ZV|\0<>9UK8AmWqQR0^A=CD9C\LS9<S36K27]`^KS9 ‖Aqk‖

A=9U02LNR9C[7λn9AM`^KS9_36/X?JQR0gN19CDh19U02?J9]9U7:Ak3EY60SI@/4Y6C=9mL89]AD/1K8u ∣

λn−1

λn

7:Yαn−1 6= 0

Z l%$ p " "+ $ c+]+ L"+%c !")&(+]"+

Q1K2Ck3E9@7kWX WX9U7nC[/4Yc7:QR027Ul^3 - Y~ADIC[/OA=Y6Q10Axk+1 = qk

qk+1 =xk+1

‖xk+1‖/NR9U? ‖q0‖ = 1lg9JA

q00y- 9U7:AT<>/17GQ1C=A=wSQRh1Q10>/43>o

v1 l8?JQ10gNR9CDh19;NR9C[7(3c/zL8Y6C=9@?BADYEQR0aL8KaNR9U?JA=9K2CG<SCDQ1<2C=9]/17D7:Q8?JY6Ioq3c/q<S3EK>7k<>9A=YEA=9mNO/4369K2Ck<SC=QR<SCD91Z l j" N& "+"1Zkxk??I36IC[/OA=Y6Q10<2/1C KPA+QWJOQT@(V3c/pW/4A=CDY6?9

A + αI/&3E9@7_WX WX9U7mNR9U?JA=9K2CD7]<SCDQ1<SCD9U7_`^KS9

A9JAz7=9U7NO/4369KSC[7+<SCDQ1<2C=9@7m7:QR0RAqL8IU?U/436I9U7zL89e3c/`RK>/40^A=YEA=I

αZ,!/rWXIJADwSQ8L89pL89@7+<SK2Y67D7=/102?J9@7mYEA=ICDI9@7mY60^NR9C[7:9@7

?JQR0^NR9CDh19XLV- /4K8A[/40^Am<S3EK>7mNgY~AD9e`RK293E9@7mCD/1<S<fQ1C=AD7 (λ1

λ2

)k A=9U02L890^A+C[/4<SYcL89UWq9U0^A_NR9C[7-7SZ#|\0/L8QR02?YE0^ADIC JA]o?9`^KS9 ‖λ1‖

7=Q1YEAM3E9z<S3EK>7;<SCDQ8?[wS9+<fQR7D7:Y6jS369zL89472l29JA]L89z<S36K27Ul23c/WXIJADwSQ8L897=9C[/LV- /4K8A[/40^A<S36K27nCD/1<SYcL89]`^KS9_3 - I@?/1C:An90^A=CD9_3E9@7nL89K8up<S3EK>7n<>9A=YEA=9U7nNO/13E9UKSCD7T<SCDQ1<2C=9@7G7=9m?JCD9K27=91Z8,!/zA=9@?[wS0SYc`^KS9mL8KL8IU?U/43c/4hR9;?JQ10>7:Yc7FAD9ML8QR02?;omCD9WX<S3c/1?9CA<>/4C

A+αIl^/N19@?

α ≈ −λ1ZO.3EK27G3 - 9@7FADYEW/OADYEQR0eL89

λ17=9C[/<SCDIU?JYc7=91lV<S36K27m3c/?JQR0^NR9CDh19U02?J9q7:9UCD/&C[/4<SYcL891Zy*Q1KSA=9JQRY67UlVY63P/4K8Az`^KS9

α 6= −λ1<fQ1KSCzINgY~AD9C+`^KS93c/W/OA=CDYc?J9_0S9+L89UN^Y69020S9+7:Y60ShRKS3EY6iCD9

8Zk!9@?[wS0SYc`^KS9;L89 PQB8MOSVU(O36/_WXIJADwSQ8L89;L89@7<SKSYc7=7D/402?9U7YEA=IUC=IU9U7<f9K8A A=CD9TIA=9U02L8KS9k<fQ1KSCP<f9CDWX9JA=A=CD93E9+?U/43c?JKS3yLS9_A=Q1KSA=9U7n369U7;N/13E9UKSC[7G<2C=QR<SC=9@7nLV- K20S9mW/OA=CDYc?J9m7:5gWXIJADC=Yc`^KS91Z & KS<S<fQR7=Q1027n9U0&9f9AM?/136?KS3EIU936/e<S3EK>7kh1C[/402LS9mN/13E9UKSCk<SCDQ1<SCD9λn

/1YE0>7:Y!`^Ky- KS0NR9U?BAD9KSCk<2C=QR<SC=9+/R7=7=Q8?JY6Ivn Z & Q1YEA Pn

3c/X3c/XWX/4A=CDY6?9L89k<SCDQ F9@?BA=Y6Q10qQ1C=A=w2Q1h1QR02/4369G7=KSC(3 - w^5g<f9CD<S3c/40(vn)⊥

ZR,!/]W/OA=CDYc?J9PnA

<>Q^7=7=iUL89n369U7PWX UWq9@7N19@?BA=9UKSC[7<SCDQ1<SCD9U7P`^KS9A9A.3E9@7PWX WX9U7(N/13E9UKSC[7<SC=QR<SCD9U7o_3H- 9JuS?9<8ADYEQR0XLS9

λn`^KSYS9@7FA(CD9WX<S3c/1?I9n<2/4C72Z1,(- /4<Sb<S36Y6?U/OA=Y6Q10pL89;3c/mWXIJA=w2QgLS9\L89@7(<SKSYc7D7=/102?J9@[email protected]

PnA<f9CDWX9JA:ADCD/zL8Q102?ML89\?U/43c?JKS369C(3c/+L89UK8u8YEiUWX9<S36K27zhRCD/102L89eNO/13E9UKSCz<SCDQ1<2C=9eL89

AZ 9JA=A=9eAD9U?[wS02Y6`^KS9Rl#L8YEA=9&L89pLSI>/4A=Y6Q10yl*<>9UC=WX9JAA=wSIUQ1CDY6`^KS9UWX90^AL89e?/136?KS369C_A=QRK8A=9@7_3E9@7_NO/4369KSC[7_<SCDQ1<SCD9U7_L89

AZy(363E99@7FA_A=QRK8A=9Q1Yc7m0gKSWXICDY6`^KS9UWX90^AmYE027:AD/1jS369X7D/40>7<SCDIU?/1K8A=Y6Q10>7l>9JA_Q1036KSY<SC=IICD9C[/Xh1IU0SIC[/4369WX90^AM3c/WXIJADwSQ8L89XL89U7\<SKSYc7=7D/40>?J9U7Mh1CDQ1KS<fI9@7k<SCDIU7=90^A=IU9?JYEb /1<SCDiU7UZ

%&)J!*+&'Gk 1+ #"+%pkk T T p"$& 0|\0m<f9K8A#h1IU0SIC[/436Y67=9CV3c/nWXIJADwSQ8L89.L89@7y<SKSYc7D7=/102?J9@7VY~ADICDI9@7yo;L89@7yYEA=IC[/OADYEQR027VL8YEA=9@7!hRC=QRKS<>IU9U7UlB<>9UC=WX9A:AD/10^ALy- YcL890^ADY29UC+369U7mNR9U?BAD9KSC[7_<SCDQ1<SCD9U7_/R7=7=Q8?JY6IU7m/4KSu

p<S36K27mh1C[/40>L89U7_NO/4369K2CD7_<SCDQ1<2C=9@7ZVx ?[w2/R`^KS9XY~ADIC[/OA=Y6Q10!l

Page 64: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

Q10 /4<S<S36Yc`RK2936/ A=C[/4027:Q1CDW/OA=Y6Q10

Ao ?[w2/1?KS0 L89U7pN19@?BAD9KSC[7Q1C=A=w2Q10SQRC=WXIU7Ul(9AaQ10 QRC:ADwSQ1hRQ102/13EYc7:9 ! <2/4C3]C[/4Wqb & ?[wSWXY6LgAkQRKMQ1K27=9w2Q13cL89C@l q ?[w2/1<SY~ADC=9 $.369+7=5g7:A=iUWX9]CDIU7=KS3EAD/10RA@Z

G `)& N S p"+ $ +*+ L"+%`)&)"+& QRY~A q(k)

1 · · · q(k)p

KS07=587FADiWX9mL89pNR9U?JA=9K2CD7nQRC:[email protected]=9U0^Koq3 - YEA=IUCD/4A=Y6Q10

k ?JQ10>7FADC=KSY6CD9\369+7:587:A=iUWq9 Aq

(k)1 · · ·Aq

(k)p

?/43c?JK23E9UCk3E9@7k`RK2Q4A=Y690^A[7kL89 /5g3E9UYEhRw λi = q(k)i

TAq

(k)i , 1 ≤ i ≤ p

A=9U7:A=9UCk36/X?Q10gN19UC=hR902?9 max1≤i≤p

‖Aq(k)i − λiq

(k)i ‖ < Tol

Q1C=A=wSQRh1Q10>/436Y67=9C Aq(k)1 · · ·Aq

(k)p → q(k+1)

1 · · · q(k+1)p

G p p"

,!/zWXIJA=w2QgLS9 ?Q1CDC=9@7:<fQ10>L/4Ka?U/17p = n

87:QRY~AQk

3c/+W/OADC=Yc?J9 ! n×n$(L8QR0^AG369U7n?JQR3EQR0S0S9U7G7:QR0^AG369U7

N19@?BAD9KSC[7kQ1C=A=w2Q1h1QR02/4KSuq(k)i

9JA]7:QRY~AAk = QT

k AQk3c/C=9U<SCDIU7=90^AD/4A=Y6Q10rL89

A7=KSC;3c/Xj2/17=9zL89@7 q(k)

i Z|\0<f9K8AMI@?JCDYECD9_/436Q1C[7 ! 3]CD/1Wqb & ?[w2WqYcLgA $

AQk = Qk+1Rk+1Q+,Rk+1

9@7FAkADC=Yc/40ShRKS3c/4Y6C=9_7:K2<>IUC=Y69KSCD91ZgPA]/4Y6027:YAk = QT

k Qk+1Rk+1 = QRk+1

Ak+1 = QTk+1AQk+1 = Rk+1Q

Tk Qk+1 = Rk+1Q

(0 CDIU7=KSWXI1lQ9U7:A36/W/OADC=Yc?J9&Q1C=A=wSQRh1QR02/4369&`^KSYk<f9CDWX9JAaL89rADC=Yc/402h1KS3c/4CDY67=9C

Ak9AeQ10sIU?JCDYEA

Qk+19U0Y60^NR9C[7=/10^AT369m<SCDQgLSKSY~A

QRZS,!/qWXIJADwSQ8L89 <SCD902L&/436Q1C[7T36/Q1CDWX9_<2/4C=A=Yc?JKS36Y6iCD9WX90^A;7:Y6WX<S369+7:KSY6NO/40^A=9

A0 = A Ak = QR ! <2/1C3]CD/1Wb & ?[wSWXYcLgAk<2/4Ck9u89WX<S369%$ Ak+1 = RQ A=9U7:AM7=KSCk3E9_<S36K27khRCD/102LpI36IWX90^Ak0SQ10L8Yc/4hRQ102/13|\j27=9CDN1QR027n`RK29

Ak+1 = QT AkQl2?J9m`^KSYyY6WX<S36Y6`^KS9z`^KS9_369U7kW/OADC=Yc?J9@7k7:QR0^ATADQ1K8AD9U7;7=9WjS3c/4jS369U7n9AM`RK29m36/7=KSY~AD9L89U7\W/OA=CDYc?J9U7

Q?Q10gN19UC=hR9+N19UCD7\36/aW/OA=CDYc?J9XL89U7]N19@?BAD9KSC[7M<2C=QR<SC=9@7Zf,!/pW/OA=CDYc?J9

Ak?Q10gN19UC=hR9+N19UCD7KS0S9mW/4A=CDY6?9_LSY6/1h1Q10>/4369\QD,p369U7kNO/13E9UKSCD7T<SCDQ1<2C=9@7T7=Q10^AkC[/40ShRI9U7nLS/1027n3 - QRCDL8CD9mL8IU?C=QRY67D7D/40^A ! *Y6h1K2C=9 Z +$JZ,y9@7n<>9UC:QRC=W/102?J9@7T7=Q10^A;?JQR027=Y6L8IUCD/1jS369WX90^Ak/4WXI36YEQRC=IU9U7n7=YyQ10&YE0^ADih1CD9mL89KSuaWXQgLSY>?U/OADYEQR027

PA+Q JNQT@ L89a3c/WX/4A=CDY6?9a9U0 <SCD902/10RAX?JQRWqWX9&/4<S<SCDQu8YEW/4A=Y6Q10 L89&36/<S36K27q<>9A=YEA=9&N/13E9UKSCq<SC=QR<SCD93H- I36IWX90^A ! n× n$nL89

Ak A=C[/4027:Q1CDW/OA=Y6Q10a<SC=I@/43c/4jS369\LS9 A9U0aKS0S9_W/OADC=Yc?J9 BC:M MOQVSTUKQWJN@fZ8,y9@7GW/4A=CDY6?9U7

AkC=9@7FAD90^AnA=CDY6L8Yc/Obh1Q10>/4369U7T9JA;3 - QRC:ADwSQ1hRQ102/13EYc7D/OA=Y6Q10e7U- 9J9U?BADKS9m90

O(n)2Q1<27UZ

X en 'G)J'Tk @DC:ADMNA+@ /1[2 "RZ & Q1YEA

a9JA

bL89UK8uCDI9U367z9JA

DKS0S9pWX/4A=CDY6?9a?/1C=CDI9RZQR0^A=CD9Cz`^KS9&7:Y

λ9@7FAN/13E9UKSC<SCDQ1<SCD9mL89

Dl2/13EQRCD7

aλ + b9@7FA;KS029mN/13E9UKSCn<SCDQ1<SCD9mL89

aD + bIl8QD,

I9@7FAk3c/XW/OA=CDYc?J9]Y6L89U0^A=YEA=I1Z8Z & QRY~A

D36/XW/4A=CDY6?9_?U/4CDC=IU9MADC=YcL8Yc/4hRQ102/13E9]L8I 202YE9m<2/1C

dii = 0, i = 1, . . . , n

di,i+1 = di+1,i = 1, i = 1, . . . , n− 1

dij = 0,/4Y63E369K2CD7

.

Page 65: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W -KK ^

Z L8Yc/4hRQ102/13EYc7D/OA=Y6Q10&LV- KS0S9mW/OADC=Yc?J9 × <2/1C LV- /4<2C=i@7 5 9

QRKSCj = 1, . . . , n

lS7:QRY~Ax

(j) 369mN19@?BAD9KSC;L89 Rn L8Q10^Ak3c/ i iWX9_?JQ1WX<fQR7D/40^A=9_9@7FA

x(j)i = sin 2i

n + 1.

Q10^A=CD9Ck`^KS9x

(j) 9U7:AkKS0rN19@?BA=9UKSCk<SCDQ1<SCD9_L89 D/R7=7=Q8?JY6I9moq36/qNO/4369K2Ck<SC=QR<SCD9

λj = 2 cos2jπ

n + 1.

2Z .0rLSIUL8KSY6CD9]369U7;N19U?JA=9UKSCD7T<SCDQ1<2C=9@7G9A;3E9@7nNO/4369KSC[7n<SCDQ1<SCD9U7nL89m3c/qWX/4A=CDY6?94× 4

7:K2YENO/40^AD9

A =

2 −1 0 0−1 2 −1 0

0 −1 2 −10 0 −1 2

@C:A+MNAD@ /)1H MNB8@ K@M SVUKQWA+ADM|\0?JQ10>7:YcL8iCD9]36/X7=KSY~AD91l1l2l3l5l8l13lVZ6Z6Z2L8I >0SYE9m<>/4C

u0 = 1, u1 = 1,

uk = uk−1 + uk−2, k ≥ 2.

/136?KS369CkKS0S9_NO/4369K2C;/4<S<SCDQ8?[wSI9_L89uklS90rK8ADYE36Yc7=/10RAk369U7k<2KSY67D7D/402?9U7TY~ADICDI9@7Z

@C:A+MNAD@ /)1 & Q1YEAa9JA

bL89K8upN19U?JA=9UKSCD7T0SQR0&?Q136YE02IU/4Y6CD9U7nL89

Rn ZSd\IJA=9UC=WXY60S9Ck369U7nNR9U?JA=9K2CD7T<SCDQ1<SCD9U7T9A369U7nNO/13E9UKSCD7n<2C=QR<SC=9@7nL89m36/qW/OADC=Yc?J9n× n

A = aaT + bbT .

Page 66: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

+ @DC:ADMNA+@ /1 SVU @DC T@U AD@ K@ JNQIF PB S @ K@QWA+S KM & Q1YEA

A′ = A + El23c/7:QRWXWq9+LS9zL89KSuW/OADC=Yc?J9@7k7:5gWXIJADC=Yc`^KS9U7UZS,y9@7nN/13E9UKSC[7n<SC=QR<SCD9U7nL89m?J9@7nA=CDQ1Yc7TWX/4A=CDY6?9U7k7=Q10^Ak0SQ4ADI9@7

λ′1 ≥ λ′

2 ≥ · · · ≥ λ′n, λ1 ≥ λ2 ≥ · · · ≥ λn, µ1 ≥ µ2 ≥ · · · ≥ µn.

"1Z QR0RADC=9UCn3E9@7k<SCDQ1<SCDYEIA=I@7T7=KSY6NO/40^A=9@7n<>QRKSCnA=QRK8Ai = 1, . . . , n

M λi + µn ≤ λ′

i ≤ λi + µ1 MOM |λ′

i − λi| ≤‖ E ‖ lS`^KS936369+`RK29_7=Q1YEA;3c/02Q1CDWq9_W/OADC=Yc?JY6936369 ‖ · ‖ Z8Z & QRY~AE(k) = A(k) − diag(a

(k)ii )

lSQD,&369U7A(k) 7=Q10^A;369U7kW/OADC=Yc?J9@7n90ShR902LSC=IU9U7n<2/1Ck36/XWXIJADwSQ8L89zL89

^/1?JQRjSY ZV(0 K8A=Y636Y67D/40^A+3c/p02Q1CDWq9 ‖ E ‖F = (ADCD/R?J9

(ET E))1/2 lyWXQ10^A=CD9Cm`^KS9 E(k) A=90>LNR9C[7_36/&W/OADC=Yc?J90gKS363E9RlS3EQRCD7D`^KS9k 7−→ ∞ ZSZ (0L8I@L8KSY6C=9m369]A=wSIUQ1CDiWX9mL89+?JQR0^NR9CDh19U02?J9]L89m36/qWXIJADwSQ8L89+L89^/1?JQRjSY l8YHZ 91Z

M a(k)ij −→ 0

l8<fQ1KSCi 6= j

MOM?[w2/1`^KS9a(k)ii −→ λi

lSQ+,λi9U7:A;KS0S9mNO/4369KSCn<SCDQ1<2C=9mL89

AZ

@DC:ADMNA+@ /1f/ & KS<S<fQR7=Q1027k?U/43c?JKS36I9m3c/q<S3EK>7;h1C[/40>L89\NO/13E9UKSC;<SCDQ1<SCD9λ1Ly- KS029mWX/4A=CDY6?9+7:5gWXIJADC=Yc`^KS9

A9JA369mN19U?JA=9UKSCk<SCDQ1<SCD9]/R7=7=Q8?JY6I

v1Z

"1Z KS9U3E369_9U7:A;36/qW/OADC=Yc?J9P1L89m<SCDQ F9U?JA=Y6Q10&Q1C=A=wSQRh1QR02/4369\7=KSC

v⊥1Z8Z QR0RADC=9UC]`^KS9

P1A/p3E9@7_WX WX9U7_NR9U?BAD9KSC[7]<SCDQ1<SCD9U7]`^KS9

Alyo&3H- 9JuS?J9U<8A=Y6Q10L89

v1Z KS9363E9@7_7:QR0RA369U7]NO/4369KSC[7M<2C=QR<SC=9@7ML89

P1AZV.0L8I@L8KSY6C=9qKS0S9q<SCD9WXYEiUC=9WXIJA=w2QgLS9qL89X?/136?KS3L89U7]<S36K27\h1C[/402L89@7MNO/13E9UKSCD7<SCDQ1<SCD9U7nL89

AZ

Page 67: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

o~ m _ _

a ! #"$&%&'(*)"+l u S

TY690 `^KS9X3E9@7_0SQ4ADYEQR027mL8INR936Q1<S<fI9@7_LS/4027m?9e?[w2/4<2Y~ADC=9X7:QRYE9U0^A]ADC=i@7_h1IU0SIC[/4369U7Ul>QR00S9?JQ10>7:YcL8ICD9C[/eYc?JY`^KS9_369+?/R7kL89U7F QWBC:MNAD@=X=8RF PBC:M HK@=1Z,P]-KUKMLB%MNSTU)1[2 QBC:MOA+@ KP]-KUKMN@?ST=%MLB8M @ KY5<;b=?3@BA3CD5 Cm=@c@DH5

AM*5kMOA_;s5FK(EAJ8K

n5FEI? "! # $T

%& R'$()$*`TgEcAN5:?'Em5:7^f5F;s5mKL? EA (∀x ∈ R

n, x 6= 0) xT Ax > 0

,!Q1C[7=`^KS9M3H- YE02IhR/13EYEA=I\9U7:AG3c/4CDh19Rl1Q10a<2/4CD3E9]L89MW/4A=CDY6?9=%@F M KP]-KUKMN@X? SV=%MLB%M @fZ^,(- YE0SIUhR/13EYEA=I;Y60gN19UCD7=9 ! < QRK≤ $G<f9CDWX9JAML89+L8I20SY6Ck3E9@7k0SQ1A=Y6Q1027kL89_W/OADC=Yc?J9@7 P]-U MO@=>UKP VQWB8M @=_9JA=%@F M KP]-KUKMN@= UKP VQB%M @=RZl ,+" $ "+"1Z- m/.10325476@sci98".[i/:<;6.s=214> 7=Q1YEA f : R

n×Rn → R

K20S9.QRC=WX9 KMOJNMOU PQWMNC @]L8I20SY69G<2/1Cf(x, y) = xT Ay

lQ+,A9@7FAnKS0S9]W/OA=CDYc?J9 ! n×n

$JZ8,yQ1C[7D`RK29A9@7FAn7:5gWXIJADC=Yc`^KS91l^Q10p3EK2Y/R7=7=Q8?JY69;3c/+QR02?BADYEQR0

qlg/4<S<f936I9

S C:F @? HKQ C QWB8M HK@fl>L89Rn LS/40>7 R

l2L8I >0SYE9m<>/4C(∀x ∈ R

n) q(x) = xT Ax

& [email protected] 20SY69k<fQR7=Y~ADYENR91l

q9U7:A.o_N/13E9UKSC[7<>Q^7:YEA=Y6N19@79JA(9Ju8wSY6jf9;L8Q102?nK20qK20SY6`^KS9k<fQ1Y60^AGL89kWXY60SYEWKSW90

x = 0lSQ+,&936369_7U- /40S0gKS3691Z`P/4Ck9u89WX<S3691l2?1- 9@7FAk369+?/17kLS9]3c/qQ10>?BA=Y6Q10r?/1C=CDI_L89m36/q0SQRC=WX9

q(x) = ‖x‖2L8Q10>?m36/qW/OADC=Yc?J9mYcL89U0RADY~ADI

I9@7FA\LSI 20SY69m<fQR7=Y~ADYENR91Zf|\0rN19UC=C[/XLS/40>7n36/7=KSYEA=9z`^KS9

q9U7:A;KS0S9_QR02?BADYEQR0

A+STU @]@fZ& Q1YEAMLS/1027R2 36/QR02?BADYEQR0

q(x) = x21 + 2x2

2 + 2x1x2 = xT Ax,/NR9U?

A =

(

1 11 2

)

|\0pQRj27=9CDN19M7=KSCk?J9AT9ug9UWX<S3E9_`^KS9]A=QRK8A=9\Q1CDWX9]`^K2/RL8CD/4A=Yc`^KS9q(x) =

i

j

qijxixj<f9K8A;7U- I@?JCDYECD9

L89]WX/10SY6iCD9MKS0SYc`^KS9_7:QRK27.3c/zQ1CDWq9xT Ax

lS/N19U?A7=5^WXIA=CDY6`^KS9RZ 3V7=K<XAnLS9\<2/1C:A[/4hR9C(IUhR/4369WX9U0RAT3E9

Page 68: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

O ) K 4 ! K A=9CDWX9m?JCDQ1Yc7:I

qijxixj/N19@?

aij = aji =qij

2 , i 6= jZ|\0p<f9K8A\Wq9A:ADC=9

q7:QRK27n36/QRC=WX9

q(x) = (x1 + x2)2 + x2

2 ≥ 0

`RK2YR0S9T7- /4020^K23E9T`RK!- 9U0 ! 72l 7$BZdMQR02?(3c/nQ1CDWX9 q9@7FAL8I 202YE9G<>Q^7:YEA=Y6N19T/4Y6027:Yg`^KS9.3c/kW/OADC=Yc?J9

A/17D7=Qg?YEIU91Z

SZ i/:<.1;_q 2 : .1;8";_i mgivo 2 892Ao 2 2 0Kego 238 254 . 214 4.m".1:<4 8vs >

A =

2 −1

−1

Z Z Z−1

−1 2

⇒ q(x) =

n∑

i=1

2x2i − 2

n−1∑

i=1

xixi+1.

|\CUlq<>9UK8A JADC=9_WXYc7:9+7=Q1K27T3c/qQ1CDWX9_Ly- KS029+7:QRWqWX9mL89m?/1C=CDIU7

q(x) = x21 + (x1 − x2)

2 + · · ·+ (xn−1 − xn)2 + x2n

36/rW/OADC=Yc?J9XIAD/40^A+0SQR0 7:Y60Sh1K23EY6iCD91lV9U3E3699U7:AzL8I >0SYE9a<>Q^7:YEA=Y6N19RZ*|\0CD9JADC=QRKSN19XYc?JY./1K ?JQg9 <e?JY690^Am<SCDiU73H- 9Ju8<SCD9U7D7:Y6Q10&L89m3 - IU0S9CDh1Y69\WXI@?/402Y6`^KS9mL8K7=5g7:A=iUWX9+L89U7nCD9U7D7:QRC:A[7Z2Z m/; 8/.)214Kqri".1.[h54 @7=Q1YEA B

KS0S9.W/4A=CDY6?9 ! p×n$A=9U3E369G`^KS9

rang(B) = nZ@xM3EQRCD7!36/;WX/4A=CDY6?9

A = BT B9U7:A\L8I >0SYE9z<>Q^7:YEA=Y6N19RZf|\0rC=9A=CDQ1KSNR9]3c/W/OADC=Yc?J9+LSK7:587:A=iWX9+L89@7kIU`^K2/4A=Y6Q1027;0SQ1CDW/4369U7 !q ?[w2/1<SYEA=CD9 $BZ & YB0y- 9U7:A;<2/17kL89mC[/402h<S369Y60yl

A9@7FA;7=9KS369WX90^AM7=9WXYEb L8I20SY69_<>Q^7:YEA=Y6N19RZ

e[ m +'.*n #)!_*)"+ $&k _! )J'Tk $&&&)Jk "+!)[#) +kl +]" +. +m '& " p" q&&+

Pu89WX<S369q(x) = 10x2

1 + 5x22 + 2x2

3 + 4x1x2 − 2x2x3 + 6x3x1

= (x1 + 2x2)2 + (x2 − x3)

2 + (3x1 + x3)2

= xT Ax/NR9U?

A =

10 2 32 5 −13 −1 2

d\Q102?A9@7FA;7=9WXY~b L8I20SY69m<>Q^7:YEA=Y6N19RZ

" BB%@U B%MNSTU(Q1K2C(WXQ10^A=CD9C.`^KS9M36/_W/OADC=Yc?J9M9U7:A.L8I 202YE9\<>Q^7:YEA=Y6N19Rl4YE32/4K8AGWqQR0^A=CD9C.`^Ky- 9363E9M9U7:AGL89;<S36K270SQR0r7=Y60Sh1KS36Y6iCD9 ! ?J9+`^KSYy9@7FAk369+?/17k?Y~b L89@7=7=K27 $BZl "&(+ $& S$&"+$6 S+* !"+ PS C F @?)1[2 KY5 ;b=`?3@BA[CF5gE;sH8?[@cA[h`7]5<M*HKA5vNJEAP?3APt85vNJEcEFe M`5<M*5DE tD=^f5:7@BEvL@DJDvL@D5FEEI?3@BA3C ?Q5m;s5FKY?vqJ:EcA[?[A[t:5FE

wyxz|~L]~| "RZ & KS<S<fQR7=Q1027

A7=5^WXIA=CDY6`^KS9nL8I >0SYE9n<fQR7=Y~ADYENR91Z & QRY~A

λiKS0S9nNO/4369KSC<SCDQ1<SCD9G/R7=7=Q8?JY6I9n/4KN19U?JA=9UKSC<SC=QR<SCD9

xiZf|\0&/XL8QR02?

Axi = λixil89JA

xTi Axi = λix

Ti xi = λi‖xi‖2 > 0

Page 69: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

K 4 ' K 4 ! K 8Z IU?YE<2C=Q8`^KS9WX9U0RA@l17=Y8A=QRK8A=9@7P3E9@7PNO/4369KSC[7<SCDQ1<SCD9U77=Q10^A(<fQR7=Y~ADYENR9U7UlOQ10/]<>QRKSCPA=QRK8A(N19@?BAD9KSCP<SCDQ1<2C=9

xi

xTi Axi > 0

Z 9U7]L89CD0SY69C[7nQ1CDW/40^A\KS029zj2/R7:9+QRC:ADwSQ102Q1CDWqIU9 ![q ?[w2/1<SY~ADC=9 $BlfQ10I@?JCDY~A]<>QRKSC;ADQ1K8Ax 6= 0

xT Ax =

(

n∑

i=1

αixi

)T

A

(

n∑

i=1

αixi

)

=

n∑

i=1

α2i λi > 0

P/4Ck9u89WX<S3691lS369U7nNO/4369K2CD7n<SCDQ1<2C=9@7TLS9m36/qW/OADC=Yc?J9mL8K7=5g7:A=iUWX9_LS9U7nCD9U7D7:QRC:A[7T7=Q10^A(∀j ∈ 1 · · ·n) λi = 2− 2cos

2jπ

n + 1> 0

l%$ N& "+y+] + `&( L"+ q&&+>+ `&( p"+ $ q& & q$$6 '&` ' S "*Q1K8AD9U7z3E9@7+7:QRK27:b W/OADC=Yc?J9@7m?/1C=CDIU7m7=5gWqIA=CDY6`^KS9@7m<2/1C+CD/1<S<fQ1C=Azor36/LSY6/1h1Q10>/4369XLy- KS029eW/OA=CDYc?J9eL8I20SY69<fQR7=YEA=Y6N19_7:QR0RAnL8I >0SYE9@7T<>Q^7:YEA=Y6N19@7Zg(0&9J9JA@lg7=Q1YEA

AkK20S9]A=936369]7=Q1K27:b WX/4A=CDY6?91l^?JQ10>7FADY~ADKSI9_L89U7

k<SC=9UWXYEiUC=9@736Y6h10S9@7*9A(?JQ136Q1020S9U7LV- KS0S9nW/OADC=Yc?J9

AL8I 202YE9n<fQR7=Y~ADYENR91Z wSQ1Yc7=Y67D7:QR027*KS0NR9U?JA=9K2C

xL8QR0^A369U7

n−kL89UC=0SY6iCD9U7?Q1WX<fQR7D/40^A=9@7T7=Q10^Ak0gKS363E9@7

xT Ax =(

xTk 0

)

(

Ak ∗∗ ∗

)(

xk

0

)

= xTk Akxk

?9m`RK2Y!L8IWXQR0RADC=9_3c/q<SC=QR<SCDYEIA=I1Z(0<2/4C=A=Yc?JK23EY69C@l2KS029?JQR02L8YEA=Y6Q10'UKPA+@=%=8QWMNC:@a<fQ1KSC_`^Ky- KS0S9zW/4A=CDY6?9z7:QRY~A_L8I >0SYE9<>Q^7:YEA=Y6N199U7:A]`^KS93E9@7IU3EIUWX90^AD7kL8Yc/4hRQ102/1K8ua7=Q1Y690^Ak<>Q^7:YEA=YE7Zl N`&(L" L" ' S ",P]-KUKMLB%MNSTU)1 QBC:MOA+@ KMNQTSVUKQWJO@ STF MNUKQWU B%@ KY5!;b=?3@BA3CD5

A5DE?qM.A[?5 $PZ & #qZ`XT R ( $ (DTUjT/# ( & U $<#NZ # (FT EcA

(∀i ∈ 1 · · ·n) aii >

n∑

j=1,j 6=i

|aij |

PSVCF @?1 5DEl;b=`?3@BA[CF5FE MOAf=8dJ8K =^f5pE?[@cA[CI?5m;s5mKL?M*J:;yA_K =KL?Q5EmJ:KL?!M`H KA5FE!vNJEAP?3APt85DEwyxz|~L*~L| 6|\0pKSA=Y63EYc7:9+Ly- /1j>QRCDLp36/X7=5gWqIA=CDYE9mL89

A9A;3E9_/1Y~AM`^KS9

2|xi||xj | ≤ x2i + x2

j<fQ1K2CkIU?JCDY6C=9−∑

i6=j

aijxixj ≤∑

i<j

|aij ||xi||xj |

≤∑

i<j

|aij |x2i +

i<j

|aij |x2j

,!9]LS9CD0SYE9UCT7=9U?Q102LeWq9UWzjSCD9M<f9KSAk7U- IU?JCDY6C=9 ∑i

j 6=i

|aij |x2i

l^9AT3c/<SC=QR<SCDYEIA=I\LS9]L8QRWXYE02/102?J9]L8Yc/4h1QR02/4369Y6WX<S36Y6`^KS9+`^KS9m?J9_AD9CDWX9]9@7FAM7:A=CDYc?BA=9UWX90^AkYE08IUC=Y69K2C;o ∑

i

aiix2i

l2Ly- QD,

−∑

i6=j

aijxixj <∑

i

aiix2i

Page 70: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

7 ) K 4 ! K l[ ! + $6 S+* F+ PS C F @?)1 KY5l;>=?3@BA3CD5M*H (KAf5SvNJEAP?3APt85lEm5F@m=<?[@cAf=:K`dY7^f=:@BAaEFH5!Em=:K(ENvN5F@c; 7?P=?3AJ8K(ES=]t:5 ClM*5DENvA[t:J*?[EvqJ:EcA[?[A E wyxz|~L]~| & Q1YEA

AKS029eW/OA=CDYc?J9eL8I20SY69e<>Q^7:YEA=Y6N19RZ!*Q1K8AD9U7m369U7+7=Q1K>7Fb W/OA=CDYc?J9U7_<2C=Y602?JY6<2/13E9@7m7:QR0RAL8QR02?rL8I 202YE9@7X<>Q^7:YEA=Y6N19@7 ! L8Q10>?p02Q107:Y60Sh1K23EY6iCD9U7 $BZ 93c/YEWX<S36Y6`^KS9`^Ky- YE3k9u8Y67:A=9rKS0S9&/R?BA=QRC=Yc7D/OA=Y6Q10 A =

LDLT l/N19@? DL8Yc/4hRQ102/13E9z9JA

LA=CDY6/10Sh1K236/1YECD9mYE0SICDYE9UKSC=9RZfd\Q10>?

D = L−1A(LT )−1 9A]?[w2/R`RK29zI36IWX9U0RAL89D7U- I@?JCDY~A

xT AxlSQ+,

xT 9U7:AnK20S9m3EY6h1029_LS9L−1 ZS,y9@7k<SY6N1Q4A[7n7:QR0RA;L8QR02?_<fQR7=Y~ADY~7UZ

9+CDIU7=KS3EAD/OA@l>?9C=AD/4Y60S9UWq9U0^A\LV- KS0h1C[/40>LpY60^A=IUC/ AM<SC[/OA=Yc`^KS91l>NgYE9U0RA]L8KCDIU7=KS3~A[/OAM<23EK27\h1IU0SIC[/43y7=KSY6NO/40^A7=Y(Q10IU?C=YEA+KS0S9XQRC=WX9`^K2/1LSCD/4A=Yc`RK29q<>/4CmC[/4<S<fQ1C=A+orKS0S90SQRKSN19U3E369Xj2/17=91lyQR0C=I@/436Y67=9K20S9 A+STU VC:H @UKA+@fZ(CD90SQR027k369+?[w2/402h19WX9U0RAkL89mNO/4CDYc/4jS369

x = CylC0SQ107=YE0ShRKS36YEiUC=9RZg,*/qQ1CDWX9_`^K2/RL8C[/OA=Yc`^KS9

q(x) = xT AxL89UN^Y690^Aq(y) = yT CT ACy

Zf|\0/q/13EQRCD7Ul87=/1027kL8IUWqQR027:A=C[/OA=Y6Q10 PS C F @?)1 STMS#MNUK@C%B%MN@K@ RJ @=8B8@C CT AC

= ^f5;srm;s5KYJ:; n:@D56M*5 tD=:^587@mE'vL@DJDvL@D5FE vqJEcA[?[A[t:5FE KYHBd`=`?3APt85DEJ*7<KL7^_^5FEh`7]5 A

K2/102L]Q10m<SY6N1Q1A=9(KS029(W/OA=CDYc?J9

A7=5gWXIJA=CDYc`RK29(7=/1027V<f9CDWzK8A[/OA=Y6Q10!lQ10m3c/kWX9JA#7:QRK27V3c/nQ1CDWX9

A = LDLl/N19@?

DL8Y6/1h1QR02/4369M?JQR0^A=90>/40^A.ADQ1K27G3E9@7T<SYENRQ4A[7ZgdMQR02?4lg369U7 =%M TU @= K@= ?M SVB8=_?Q 02?JYcL890^An/N19U?M?J9KSuaL89U7NO/4369KSC[7n<SCDQ1<SCD9U7 ! WX/1Y67;<2/17n369KSC[7nNO/4369KSC[7 $BZ

l cp" q&&()" p" `&( L"& Y

A9@7FAkKS029mWX/4A=CDY6?9mL8I 202YE9m<fQR7=Y~ADYENR91l8Y63y9Ju8Yc7FAD9_KS0S9mW/OADC=Yc?J9_0SQR0&7=YE02h1KS36YEiUC=9

RAD9363E9+`^KS9

A = RT R.|\0/4<S<f936369]<>/4C=Q1Yc7R36/qC[/1?YE0S9m?U/4CDC=IU9]L89

AZ>,y9+?[wSQRY~upLS9

R0y- 9U7:An<>/17nKS0SYc`^KS9

/N19U? A = LDLT lSQ10&Q1jSA=Y690^A R = D1/2LT

7:QRY~A Q36/XW/OADC=Yc?J9mL89@7nN19@?BA=9UKSCD7n<2C=QR<SC=9@7l

QT AQ = LlSL8Q102?

R = L1/2QT

R = Q′L1/2QT /N19U?Q′ QRC:ADwSQ1hRQ102/13E9

@F QWC HK@ 132 KY5[J8KC ?3AJ8K h*7]=*MO@F=`?[A3h`7]5M`H KA5!vNJEAP?3APt85SvN5:7? r:?3@D5CDJ8K(EcA3M`Hm@DH5CDJ8;y;s5u7KY55B9?5mKEcAfJ:K M*5^f= KYJ8@B;s5u587`CB^_A[MOA5mKKY5u=IvL@DeFEC =:Kd`5m;s5FKY?GM*5 tD=@cAf=n:^5 C`\ 5DE?M*J:KC=`7OEBEcA 7KY5KYJ:@c;s5

Ry = x⇒ ‖x‖2 = yT RT Ry.

e/. T&"$p$& &"+Jkq,(- /1L2/4<8A[/OA=Y6Q10rL89m3c/XWXIJA=w2QgLS9zL8943_/4K>7=7k/1K8upW/OADC=Yc?J9@7k7:5gWXIJADC=Yc`^KS9U7ML8I 202YE9@7k<>Q^7:YEA=Y6N19@7k?JQR02L8KSYEAMoX36//1?JA=QRC=Yc7=/4A=Y6Q10&L89 wSQ1369U7g51Z Q1WXWX9XYE3(/&IA=IXQ1j27=9CDN1Iq/4K<2/4C[/4hRCD/1<SwS9+<2C=I@?JIULS90^AUlV3c/e/R?BADQ1CDY67D/OADYEQR0

A = LDLT = RT R7U- I@?JCDY~A/N19@?

RA=CDYc/40ShRKS36/1YECD9_7:KS<fICDY69KSCD91ZS,.- /13EhRQ1CDY~ADwSWX9_?Y~b L89@7=7=Q1K27TCD9WX<S3c/1?9]369U7;I36IWX90^AD7

aij<2/4Ck369U7kIU3EIUWX90^AD7

rij 9JAM/13EhRQ1CDY~ADwSWX9]LS9W/402LS9 O (n3

6

) 2QR<27ZS,!9U7nIU3EIUWq9U0^AD7;L89R7=/4A=Yc7FQR0RA;L89_<S36K27

r2ij ≤

i∑

k=1

r2ik = aii, ∀i, j = 1, . . . , n.

d\Q102?rADQ1K27a3E9@7eIU3EIUWq9U0^AD7aLS9R7=Q10^AajfQ1CD0SIU7e90 WXQ8L8KS369<2/4Cp3E9@7I36IWX9U0RA[7aL8Yc/4hRQ102/1K8u L89

AlT?9`^KSY?JQR0^A=CDYEjSK29]o36/q7:AD/4j2YE36Y~ADI_0^K2WqIUC=Yc`^KS9_L89\3c/WqIA=wSQ8L89 ! 7=/1027T36/C=9U02L8CD9MADQ4A[/4369WX90^AnYE027=90>7:Y6jS369]/1KaW/4KSNO/1Y67?JQR02L8YEA=Y6Q10S029WX90^A $JZ

Page 71: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

! K K 4 K "" J TS C:MNB F @ / IJADwSQ8L89mL89 wSQ1369U7g5iSTH C

k ∈ 1 · · ·n QWMLC:@

akk ←[

akk −k−1∑

p=1

a2kp

]1/2

iSTHCi ∈ k + 1 · · ·n QWMNC:@

aik ← 1akk

[

aik −k−1∑

p=1

aipakp

]

MNUjiSTHCMNUjiSVH C

a p"+p'.*)J"zp X%&+$p #_#)X%pk 'G"+ + ek;M0S9 SVUKAB%MNSTU H Q C:QB8M< H @&L89

Rn LS/4027 R<f9K8A\7U- IU?JCDY6C=9_7:QRK27n3c/zQRC=WX9_h1IU0SIC[/4369

f(x) = xT Ax + aT x + b

QD,A9U7:AmKS0S9XW/OADC=Yc?J9X7:5gWXIJADC=Yc`^KS91l

a ∈ Rn, b ∈ RZyxMYE0>7:Y lVKS0S9qQ10>?BA=Y6Q10`^K2/1LSCD/4A=Yc`RK29z9U7:Am3H- 9JugAD9027=YEQR0o

Rn LV- KS0s<>QR3E5g0SRWX9pL8K7:9@?JQR02L L89Uh1CDI1l#9JAQ10 <f9KSAX3c/C=9U<SC=I@7:9U0^A=9Cq?JQRWqWX9&36/7=Q1WXWX9pLV- KS0S9&Q1CDWX9`^K2/RL8C[/OA=Yc`^KS9]9JAMLV- KS0S9_QR02?BADYEQR0r/<0S9 ! ?BFZ 2h1KSCD9 Z#"%$

Z#" QR02?BADYEQR0&`^K2/RL8CD/4A=Yc`^KS9mL89 R2

lp l S Y` S+ L SS"+l"+,P]-KUKMLB%MNSTU)1 STU A+B8MNSTU A+STU @]@ K MOA[?h`7\ 7KY5 [J:KCI?[AfJ:K

ntD=:@BAo=(n:^f5DEp5FEI? & # *`TqTyEcA

∀x, x′ ∈ Rn, ∀λ ∈ [0, 1]) f(λx + (1− λ)x′) ≤ λf(x) + (1− λ)f(x′). ! Z " $

& Y!3H- YE02IhR/13EYEA=I ! Z#"%$T9U7:A;7FADC=Yc?BAD91lSQR0p<>/4CD3E9mL89]Q10>?BA=Y6Q10 =8BC:MNA+B8@F @U B ADSVU @]@>Zd\9m<S36K27Ul27:YV<fQ1K2CkKS0α > 0

l2`^KSYV0S9+L8IU<>9U02Lr`RK29+L89fQ10/

f(λx + (1− λ)x′) ≤ λf(x) + (1− λ)f(x′)− 1

2λ(1− λ)α‖x− x′‖2

QR0&LSY~AM`^KS9f9U7:A SVC B8@F @U B ADSVU @]@>Z& Y

f9@7FA;?JQR0gN19Ju89Rl^3c/Q102?JA=Y6Q10

g = −f9U7:AA+STU ADQ @>Z

Page 72: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

) K 4 ! K i C:S ? SV=%MLB%MNSTU 132 J:Af5FKY?

f1, f2M`587:9 [J:KCI?[AfJ:K(ECFJ:KLt85B9 5DE

#Df1 + f2

5FEI?!CDJ8KLt85m9 5

αf15DE?GCDJ8KLt85m9(5!vqJ*7@

α ≥ 0

Wsupf1, f2

5FEI?GCFJ:KYt:5B9 5

|\0NO/+CD9JADC=QRKSN19UCPh1IQRWXIJA=CDYc`RK29WX90^A.<2/1C(3 - IA=K2L89_L89@7.?JQRKSCDj>[email protected]\0SYENR9U/1KXLS9\?J9@7PQR02?BADYEQR027.369;CDIU7=KS3EAD/OAQ10>LS/4WX90^A[/43y7=KSYENO/10RA

PS C F @?)1f/ KY5 [J8KC ?3AJ8Kh`7.=`MO@F=`?3A[h`7]5b5DE?gCFJ8KLt:5B9 5yEcAl5:?EF587^5m;s5FKY?EAp5m^a^5s5FEI?6=EBEmJCcAH5 7KY5;b=`?[@cA[CF5 EF5m;yA M`H KA5bvqJ:EcAP?3APt85 A^o= ;b=?3@BA3CD5 5FEI?>M*HKA5bvqJ:EcA[?[A[t:5 =:^J8@mE>^o= [J8KC ?3AJ8K 5DE? [J8@ ?Q5m;s5FKY?CFJ:KLt85B9 5

wyxz|~L]~| 37:KW<qAmL89qWXQ10^A=CD9C_`^KS9z3c/QRC=WX9X`RK>/1L8C[/OADY6`^KS9f(x) = xT Ax

/N19@?AL8I 202YE9<fQR7=Y~ADYENR9e9U7:AKS029eQ10>?BA=Y6Q10 ?Q10gN19u891Z

AIJAD/10^A7:5gWXIJADC=Yc`^KS91l*9363E9a9U7:AqL8Yc/4hRQ102/13EYc7=/1jS369X7=KSC36/j>/17=9aL89a7=9U7N19@?BAD9KSC[7q<SC=QR<SCD9U7+QRC:ADwSQ102Q1CDWqI@7l#7:QRY~A

A = XDXT ZPx;<2C=i@7q?[w>/40ShR9WX90^AqLS9&j>/17=9 x = Xylf7U- I@?JCDY~A

g(y) = yT Dyl!`^KSY(9@7FAz?Q10gN19u89q?U/4Cm7:QRWXWq9eLS9XQ102?JA=Y6Q1027m?/1C=CDI9@7_o&?Qg9 <e?JY690^AD7_<fQR7=YEA=YE7Z#dMQR02?

f9@7FA?JQR0gN19Ju89RZd\9m<S3EK>7lS<fQ1K2CkKS0S9mW/OADC=Yc?J9mL8I20SY69m<>Q^7:YEA=Y6N19Rl8Q10r/

λ1‖x‖2 ≤ xT Ax ≤ λn‖x‖2

Q+,λ1 ! CD9U7=<yZ λn

$k9U7:A_36/a<S3EK>7\<f9JADY~AD9 ! C=9@7:<!Z>hRCD/102L89%$;NO/4369KSC\<SC=QR<SCD9L89 AZ |\090L8IUL8K2Y~A ! ?U/43c?JKS3c7MY60^A=9UC:bWXIUL8Yc/4Y6CD9U7n3c/4Yc7=7=IU7T90r9ug9UCD?Y6?9%$T`^KS9

f9@7FAnQRC:AD9WX90^A;?JQR0^NR9Ju89_/N19U?

α = λ1Z

'& "+ "c$p "+

PADK2L8Y6Q1027.3H- 9J9JAGLy- KS0p?[w2/40ShR9WX90^A(L89;NO/1C=Yc/4jS369U7P7=KSC(3c/m0SQRC=WX9k9UK2?J36Y6LSYE9U0S0S91Z & Q1YEABKS029;W/OA=CDYc?J9k0SQR07=YE0ShRKS36YEiUC=99JA

x = ByK20?[w2/10Sh19UWq9U0^A]L89N/1C=Yc/4j23E9RZf,!9q?U/4CDC=IzLS9z3c/a0SQRC=WX99K2?3EYcL8Y690S0S9q9U7:A\K20S9zQRC=WX9`^K2/1LSCD/4A=Yc`RK29_LSI 20SY69m<fQR7=Y~ADYENR9_L2/4027

Rn

‖x‖2 = xT x = yT BT By = ‖y‖2BQ+, ‖.‖2B 9@7FAnKS0S9\0SQ1CDWX9M9U3E36YE<SA=Yc`RK29]/R7=7=Q8?JY6I9Mo

B ! BT B9U7:ATLSI 20SY69]<>Q^7:YEA=Y6N19;9ATQR0aL8IUWqQR0^A=CD9;/1?YE369WX9U0RA`^KS9m3 - Y60SIh^/436Y~ADImL8K&A=CDY6/10Sh1369\9@7FAM?Q1027=9CDN1IU9%$JZ

@F ?KJO@ 132B =

(

2 00 1

) MN\ JA =

(

4 00 1

) 5:? ‖y‖2B = 4y21 + y2

2

5 C =:K`d5m;s5mKL?yM*5 tF=@cAf=n^f5 5DE? A[CcA 7K C =K`d`5F;b5FKL?yMN\ H8C Y5m^a^5 \ 5FK(EF5m; n^f5 M`5FE vqJ8AaKL?[E ?Q5m^oE h`7]5‖x‖2 = 1 ^f= n*J`7^5 7KAP?QH M*5:t A5mKL?p^F\ 5m^a^_A vEF5 4y2

1 + y22 = 1

K \ =vvL^f=`?[AaEBEF5m;s5mKL?6M*5y^o= EPv Lem@D5<A_K(A[?[Ao=^f5EI7A[tD=KY? ^F\ =89 5 YJ8@BA:J8KL?P=^5FEI?v@FJvNJ:@I?[AfJ:KKY5m^ √

a11 C d 7@D5

Page 73: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

! K K 4 K

Z TQ1K23E9_KS0SYEA=Im9U3E36Y6<8A=Yc`^KS9

lp%$ l S Y` S+ "c$p "+ +R2

& QRY~AA =

(

5 44 5

) 9A;36/Q10>?BA=Y6Q10`^K2/RL8CD/4A=Yc`^KS9]LS9R2 /R7=7=Q8?JY6I9

q(x) = xT Ax

QRKSC;L8Yc/4h1QR02/436Yc7:9UCAlSQ10r?/136?KS3E9m7=9U7nNO/4369K2CD7n<SCDQ1<2C=9@7G9AM7:9@7nN19@?BA=9UKSC[7n<SC=QR<SCD9U7

λ1 = 1/N19@?

q1 =

(

1/√

2

−1/√

2

)

λ1 = 9/N19@?

q2 =

(

1/√

2

1/√

2

)

& Q1YEAQ3c/ WX/4A=CDY6?9QRC:ADwSQ1hRQ102/13E9LSQ10^Ap369U7r?JQR3EQR0S0S9U7&7=Q10^Ap369U7pN19@?BA=9UKSC[7a<SCDQ1<2C=9@7ZTP9U?BADKSQ10>7&/13EQRCD7a3E9?[w>/40ShR9WX90^AkL89mNO/4CDY6/1jS3E9@7

x = QyZS3c/Q1CDWq9m`^K2/RL8CD/4A=Yc`^KS9

qL89NgY690^A;/436Q1C[7

yT QT AQy = yT Λy = y21 + 9y2

2

9A;Q10&C=9UW/4C[`RK29]L89m<23EK27k`^KS9y = QT x⇒

y1 = x1√2− x2√

2

y2 = x1√2

+ x2√2

,*/Q1CDWq9`^K2/RL8C[/OA=Yc`^KS9rYE02Y~ADY6/13E9KS029rQ1Yc7eL8Yc/4h1QR02/436Yc7:IU9&7U- I@?JCDY~AaLSQ102?7=Q1K27X3c/Q1CDWX9LV- K20S97=Q1WXWX9rLS9?U/4CDC=I@7q(x) =

(

x1√2− x2√

2

)2

+ 9

(

x1√2

+x2√

2

)2

,*/(#YEhRKSCD9 Z CD9<SCDIU7=90^AD936/vj>QRKS3E9KS0SYEA=I9363EY6<8A=Yc`^KS9q(x) = 1

`^KSY_7- Q1jSA=Y690^Apo <>/4C=A=Y6CpL8936/ jfQ1K23E99UK2?J36YcL8YE9U0S0S9_<2/1CGADCD/1027:Q1CDWX/4A=Y6Q10eQRC:ADwSQ1hRQ102/13E9Q ! CDQ4A[/OADYEQR0pL89 −π/4

$Bl8<SKSYc7k/4<2<S36/4A=Yc7=7=9WX9U0RAk7=KSYENO/10RAk369U7/4u89U7n<SCDQ1<fQ1C=A=Y6Q10S0293Vo 1√λ1

Z

Page 74: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

) K 4 ! K

Z SQ10>?BA=Y6Q10r9U3E36YE<SA=Yc`RK29+L89 R2

Page 75: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

o~

m m

X )n kz#)Jr)JJ)[*|\0 ?Q1027=Y6LSiCD9XL2/4027+?9a?[w2/1<SY~ADC=9eLS9U7_QR02?BADYEQR027z?Q10^A=Y60gKS9U7mL8I20SY69U77:KSC+A=Q1KSAzQ1K<2/1C:ADYE9eL89a3H- 9U7=<2/1?9NR9U?JA=Q1CDY693

Rn Z*|\002Q4A=9UCD/ xT y369<SC=Q8L8K2Y~A+7D?/43c/4Y6CD9zK27=KS93(LS9XLS9K8uNR9U?JA=9K2CD7

x9JA

y9A ‖ x ‖ 3c/p02Q1CDWq99UK2?J36YcL8YE9U0S0S9+L8KrNR9U?BAD9KSC

x ! ‖ x ‖= (xT x)1/2 $BZ,P]-KUKMLB%MNSTU 1[2 ,6PC:M P@ MNC @A+B8MNSTUKU @JNJO@ = M`Hm@BA[t:H5gMOAa@F58CI?[AfJ:KKY5m^a^5gMN\ 7OKY5[J8K)CI?3AJ8K f

5mKx ∈ Rn

M*=:K(E^f=sMOA_@D5 C ?3AJ8Kd ∈ Rn 5DE?q^f=<^_Aa;yA[?5 h`7.=KMs5m^a^f565m9*AoEI?Q5 M`5^m\ 5B9OvL@D5FEBEAJ8K

1

t(f(x + td)− f(x))

h`7.=KMt?5mKM t85F@BE KkKLJ*?Q5F@m=

f ′(x; d) = limt→+0

f(x + td)− f(x)

t.

|\0/_L8Q10>?n<>QRKSCt7=K<e7D/4WXWX90^A(<>9A=YEA.3 - /1<S<SCDQugY6W/OADYEQR07=KSYENO/10RAD9kL89k36/]QR02?BADYEQR0

fLS/40>736/mL8Y6CD9U?BADYEQR0

dL8Y~AD9 . ; 0 : ; 8vs .)2 0 ;2 .". 8".)2

f(x + td) = f(x) + tf ′(x; d) + tε(t) ! Z#"%$QD,

ε(t)AD902L&NR9C[77q/NR9U?

tZ

,P]-KUKMLB%MNSTU 1 B8@QWH KMPC:@U B8MOQKMOJNMNB8P KY5[J:KCI?[AfJ:Kf5FEI?lMOAP?Q5! : 2 s #"8/; $&% .)2 :<; ' o 2 J:KM.A_@F=uEAa; vL^5m;s5mKL? MOA)()Hm@D5mKL?3Af=n:^5 v)=@'^f=uE7AP?Q5 5mK x

EAL5m^a^f5 vqJ:EBEFe M*57KL5M*[email protected]_@D5 C ?3AJ8KKY5F^_^5l5mKxM]=:K E

?J*7?5FE^5FEuM.A_@D5 C ?3AJ8K(E5:?qE*\ A_^N5m9`AaEI?Q5g7K t858CI?587@ ∇f(x)M*5

Rn ?Q5m^'h*7*5

f ′(x; d) = ∇f(x)T d.

,!9mN19U?JA=9UKSC ∇f(x)9@7FAM/1<S<>9U3EI . 8/;2 : L89_#9U0 x

L8I 202Yy<2/4C

∇f(x) =

∂f(x)∂x1ZZZ

∂f(x)∂xn

.

1

Page 76: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

: % > 4 K \ K K 4 ! =%@DC QB%MNSTU & Y#Q10?[w2Q1Yc7:YEA\3c/aL8Y6CD9U?BADYEQR0 d = ei

L8936/pj2/R7:9q?/402Q10SYc`^KS91l>Q10NRQ1YEA]`^KS9f ′(x; ei) =

∂f(x)

∂xi

Z

,P.-UKMLB8MOSVU )1 STHC @=K@)U M @QWH 5FE>CFJ*7O@ n`5FEbM*5<KAPt85=`7 MN\ 7KY5 [J:KCI?[AfJ:KfEFJ8KL?6M`5FEvqJ8AaKL?[E

M*5Rn ?Q5F^aEuh`7]5

f(x) = α J α5DE?7KL5gCFJ8K E?Q=:KL?Q5

& Yf9U7:AL8YEfIUC=9U0RADY6/1jS369e90

xl*369ehRCD/1LSYE9U0RAzLS9

f90

x9U7:AzQ1C=A=wSQRh1QR02/43(o3c/?Q1KSCDj>9eLS9a0SY6N19@/4K90

x9JA<fQ1Y60RAD9mN19UCD7T3c/XC=IUh1Y6Q10aQD,p3c/qNO/4369KSC;L89

f9U7:A;<S36K27nh1C[/402LS9_`^Ky- 90

xZ

,(- 90>7:9UWzjS369Sα = x ∈ R

n | f(x) ≤ α9U7:A;KS0S9 =%@AB%MNSTU L89

fZ

6

f(x) = α

f(x) = α

f(x) < α

f(x) > α

∇f(x)

G Z#" QRKSCDj>9@7kL89m0SY6N19U/1Kp9A;h1C[/1L8Y690^A

X

Z

epi(f)

G Z <SYEhRCD/1<SwS9_9JAM7=9U?JA=Y6Q10

Page 77: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

#D 4 ! ! K

,P]-KUKMLB%MNSTU 1 ?M VC:Q? @ \ HvLA_d.@F=v L5MN\ 7KY5 [J8KC ?3AJ8Kf5DE?^[email protected]*5

Rn+1 EcA[?c7]H5=7<M*5FEBE7.EM 7 d]@m=Iv Y56M*5

fepi(f) =

(x, z) ∈ Rn+1 | f(x) ≤ z

. = @FHmd.AJ8K EcAP?7]HI5u=`7 M*5FEBEmJ`7OEM(7<d.@F=v Y55DE?'=IvvN5F^fHI5 OvqJ d]@m=Iv Y5 ,!9U7&?JQ1K2C=jf9U7pL8902YENR9U/4K 7=Q10^ApL2/4027

Rn 9JAr3H- I<2YEhRCD/1<SwS99@7FArLS/4027

Rn+1 d]/4027

Rn+1 lG369NR9U?JA=9K2C

(∇f(x),−1)L8I20SYEA;3E9-?JNQ U B8QWU V@U B+oq3 - IU<SYEhRCD/1<SwS9_/4Kr<fQ1Y60RA

(x, f(x))Z

G 0 &"! ' `"% '$p `,P]-KUKMLB%MNSTU 1f/ MNUKMNFIH F TJNS QWJ F MOU MOF HKF JOSA+Q J K =vvq5m^a^f5u;<AaKAa;s7; d]^fJnO=:^ M`5^o= [J8KC ?[AfJ:K

f7KbvqJ8AaKY?

x∗ ?Q5F^ h`7]5f(x∗) ≤ f(x), ∀x ∈ R

n. ! Z D$ ALJ8K>@D5FEI?3@D5mAaKL?^o=yCFJ8K)MOA[?[AfJ:K % y7K CF5F@I?Q=:AaK t8J8AoEcA_K =:d`5pM*5 x∗ J8Ks=<7OK>;<AaKAa;s7; ^JCm=:^ D A(^F\ AaKYHBd`=:^aA[?Hv@FH8CFH M`5mKL?Q5'5DE?LEI?3@BA3C ?Q5 J8Kv=:@B^f5F@m=M`5q;yA_K(A_; 7; E?[@cA[CI? J`7@S7K ;b=89`A_; 7; Aa^*E7 ?qn:A5mKgE@!M*5GC =K`d`5F@^5EF5mK(EM*5DEA_KLHBd*=^_AP?QHDEM]=:K E^f5DEM*H K(A[?[AfJ:K(EuCFJ:@c@D5FEQvqJ8KM]=KL?Q5FE PSVCF @ )132 STU KMLB%MNSTU UKPA+@=%=8QWMNC:@ S#S ?KB8MOF QWJNMNB8P H ?KC:@F MN@CGS C C:@ A

x∗ 5DE?<7K ;yAaKA ; 7;#^JCm=^qM*5

f MOA)(HF@F5FKY?[Ao=(n:^5 J8K =∇f(x∗) = 0. ! Z $

wyxz|~L*~L| yd\9z3c/aLSI 20SYEA=Y6Q10LV- KS0WXY60SYEWKSW36Qg?U/43 l2QR0A=Y6C=9 ∇f(x∗)T d ≥ 0l ∀d ∈ Rn Z,*/36Y60SIU/1C=YEA=ImL8Kr<SCDQ8L8KSYEAk7D?/136/1YECD9]YEWX<S36Y6`^KS9+`^KS9 ∇f(x∗) = 0

Z,*/z?JQR02L8YEA=Y6Q10 ! Z D$P0!- 9@7FATh1I02IC[/4369WX90^A(<2/R7.7=K<e7=/10RAD9M?/[email protected]>I9;IUhR/4369WX9U0RAG<2/4CGKS0eW/Ou8YEbWKSW 36Q8?/43 ! 91Z h2Z f(x) = −x2 /4Ka<fQ1Y60^A 0

$GQ1KpKS0p<fQ1Y60^A;LV- Y60 29u8YEQR0 ! 91Z h2Z f(x, y) = xy/1Kp<fQ1Y60^A

(0, 0)$JZ,!9U7\<>QRYE0^AD7\903E9@7=`^KS9U367M3E9h1C[/1L8Y690^A\LV- KS0S9zQ102?JA=Y6Q107U- /40S0gKS369z7=Q10^A\/4<2<>9U3EI@7! "; :4 41: : ; ; . 214 LS93c/Q102?JA=Y6Q10yZ 3V<>9UK8A\7U- /1h1Y6CnLS9]WXY60SY6WX/2l2L89mW/Ou8Y6W/QRK&LS9m<>QRYE0^A:b 7=9363E9@7Z

G l S Y` S+ p"7+ m S+ &"! ' "+;M0S9MWKS3~ADY6/1<S<S36Y6?U/OADYEQR0q9@7FAGKS0N19@?BA=9UKSC.L89;Q102?JA=Y6Q100SQ1A=I

f(x) = (f1(x), . . . , fp(x))T Z & KS<S<fQR7=Q1027.3E9@7fiL8YEfIUC=9U0^A=Yc/4jS369U713c/mW/OA=CDYc?J9

p×nLSQ10^[email protected]=Q10^A.369U7Gh1C[/1L8Y690^A[7(L89U7.Q10>?BA=Y6Q10>7

fi9U7:AG369 ^/1?Q1jSY690LS9

f

∇f(x) =

∇f1(x)T

∇f2(x)TZZZ∇fp(x)T

.

;M0S9zQR02?BADYEQR0f9U7:A]L8YEA=9XL89UK8urQ1Yc7\L8YEICD90^A=Yc/4jS369+9U0

x7=Y*9U3E369z9@7FA_L8YEfIUC=9U0RADY6/1jS369+90

x9JA_7:Y?[w2/R`RK29?Q1WX<fQR7D/40^A=9;L8KhRCD/RL8Y690^AP9U7:A.KS0S9kQR02?BADYEQR0eL8YEfIUC=9U0RADY6/1jS369k90

xZ8|\0<>9UK8AT/13EQRCD7PLSI 20SY6C(369^/1?JQRjSY690L89;3c/WKS3EA=Yc/4<S<S36Yc?/OADYEQR0

F (x) = ∇f(x)Z,!/W/OADC=Yc?J9+Q1CDWqIU9m<2/4C\3E9@7;NR9U?BAD9KSC[7khRCD/RL8YE9U0^AD7 ∇Fi(x)

9U7:A\/4<S<f936I9 214 41;2 Q1KW/4A=CDY6?9 ;9U7D7=YE9U0S0S9zL89 f

l>0SQ4ADI9 ∇2f(x)Zf,!9 ;9U7D7=YE9U09U7:AMKS0S9+W/4A=CDY6?9+?/4CDCDI91lS7=5gWXIJA=CDYc`RK29

LSQ10^Ak3H- I36IWX90^Ahij

9U7:A;3c/qLSICDYENRI9_7:9@?JQ10>L89 ∂2f∂xi∂xj

(x)Zf|\0/XL8Q102?

∇2f(x) =

(

∂2f

∂xi∂xj(x)

)

i,j=1,...,n

.

|\0<f9K8AaL8I20SY6Ce?JQRWqWX9r<SCDIU?IUL89UWqWX9U0RA3 - /1<S<SCDQu8YEW/OADYEQR0 L8K7=9U?JQR02L Q1C[L8C=9rLS/1027X36/L8Y6CD9U?BADYEQR0dLy- KS029_Q102?JA=Y6Q10

f9U0

xQD,

f9U7:AkLS9K8upQ1Yc7kL8Y~ICD90^ADY6/1jS3E9

f(x + td) = f(x) + t∇f(x)T d +t2

2dT∇2f(x)d + t2ε(t). ! Z $

Page 78: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

: % > 4 K \ K K 4 ! PS C F @ 1 A

x∗ 5DE?g7K ;yA_K(A_; 7; ^fJCm=^pM*5f M*5:7:9 [J8AoE>MOA)(HF@F5FKL?3Af=n:^5 J8K= ∇f(x∗) = 0

58?∇2f(x∗)

5FEI?'Em5F;yA M*H (KAf5GvqJ:EcA[?[A[t:5 wyxz|~L]~| WXWXIUL8Yc/OAo;<2/1C:ADYEC#L89G3 - /4<S<SCDQu8YEW/4A=Y6Q10mL8KLS9K8u8Y6iWX9.QRCDL8CD9 ! Z $y?U/4C*3H- YE02IhR/13EYEA=I

dT∇2f(x)d ≥ 0l ∀d lSY6WX<S36Y6`^KS9+`^KS9 ∇2f(x∗)

9@7FAM7=9WXYEb LSI 20SY69_<>Q^7:YEA=Y6N19RZ,!/\?JQR02L8YEA=Y6Q10z<2C=I@?JIULS90^A=9G0y- 9@7FA<2/R7*7=K<e7=/10^A=9G?Q1WXWX9.369GWXQR0RADC=9G3H- 9Ju89UWq<23E9nL89T36/;Q102?JA=Y6Q10

f(x) = x3/4K<fQ1Y60^Ax = 0

ZP/4CT?JQ10^ADC=9Rl47:Y ∇f(x∗) = 09JA ∇2f(x∗)

9@7FATL8I 202YE9;<fQR7=YEA=Y6N191l^/436Q1C[7x∗ 9U7:A.KS0WXYE02YEWKSW36Qg?U/43y7:A=CDY6?JAML89

f ! A+STU KMLB%MNSTU =8H =8Q U B8@$BZ!'%$ " "+ & "+ " &"! N S

d]/4027M3E9@7;QRC=WKS3E9@7M?Y~b L89U7D7=Q1K27Ula ∈ Rn l A W/4A=CDY6?9z?/1C=CDI9

(n × n)lBW/4A=CDY6?9

(p × n)f9U7:A\KS029Q10>?BA=Y6Q10L89

Rn LS/40>7 RZ

f(x) = aT x + α =⇒ ∇f(x) = a

f(x) = xT Ax + aT x =⇒ ∇f(x) = (A + AT )x + a

F (x) = Bx− b =⇒ ∇F (x) = B.

& QRY~AgK20S9_Q102?JA=Y6Q10L89

Rp LS/4027

R9JA

hKS0S9_Q10>?BA=Y6Q10L89

Rn LS/1027

Rp Z2,!/qQR02?BADYEQR0r?Q1WX<fQR7=I9 f(x) =

(g h)(x) = g(h(x))/q<fQ1KSCMh1C[/1L8Y690^A

∇f(x) = ∇h(x)T∇g(h(x)).

Q1WXWX9+/4<2<S3EYc?/4A=Y6Q10rL89_36/QRC=WKS3E9m?Y~b L89U7D7=K27l8<fQ1K2Ct ∈ R

ld ∈ Rn lSQR0&/

df(x + td)

dt= dT∇f(x + td).

;;029+/4K8ADC=9m/4<2<S3EYc?/4A=Y6Q10rL89m3c/qLSICDYENO/OADYEQR0&Ly- KS029_Q102?JA=Y6Q10r?JQ1WX<fQR7=I9RZ & Yf(x) =‖ Bx− b ‖2 Q10r/

∇f(x) = 2BT (Bx− b).

q[ "+ + e)[*,P.-UKMLB8MOSVU )1 UK=8@F KJO@6ADSVU @.@ K0EmJ`7OE Q5mK Em5F; n:^5

CM`5

Rn 5DE?s7K 5mK(EF5m;kn:^5 CFJ8KLt:5B9 5 EA∀x1, x2 ∈ C ^f5EF5Bd];s5mKL? [x1, x2]

5DE?GM]=K(EC A 5 λx1 + (1− λ)x2 ∈ C ∀λ ∈ (0, 1)

@F ?KJO@ )132 5DE!J`nD58?[E'^_AaKYHD=A_@D5FE EmJ`7OE Q5FEQv)=`CF5FEGt:5 CI?J8@BAf5F^aE tD=:@BAfH:?QHDEq^_AaKYHD=A_@D5FE vqJ8^ e M.@F5DE M*5 Rn ^5FEn`J*7^5FE67OKA[?HFE=OEcEFJCcAH5DE ?J*7?5pKLJ8@B;b5 M*5

Rn P/4Ck?JQR0gN190^ADYEQR0ylgQ10r7:KS<2<>Q^7:9UCD/q`^KS9_3H- 9027=9WjS369]NgYcL89m9U7:A;?JQR0^NR9Ju891Z & Y

C19A

C27=Q10^AkL89KSu&?JQR0gN19Ju89@7L89

Rn l2/13EQRCD7TQR0&/ C1 + C2

9U7:AM?JQR0gN19Ju89 αC1

9@7FA;?JQR0gN19Ju89RlαKS0rC=IU93

C1 ∩ C29@7FAM?Q10gN19ug9RZ

,P.-UKMLB8MOSVU )1 SVUKAB%MNSTU A+STU @]@ KY5[J:KCI?[AfJ:Kf5FEI?CFJ:KLt85B9 5EI7@67K 5mK(EF5m;kn:^5gCFJ8KLt:5B9 5

CM*5

Rn EA∀x1, x2 ∈ C, f(λx1 + (1− λ)x2) ≤ λf(x1) + (1− λ)f(x2), ∀λ ∈ (0, 1).

Af5FEI?CFJ:KYt:5B9 5 g = −f

5FEI?CFJ:KCm=]t:5

Page 79: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

a b

x

z

f(x)

Z SQR02?BADYEQR0&?Q10gN19ug9

x

z

ba

Z SQR02?BADYEQR0?JQ10>?/N19

Page 80: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

7 % > 4 K \ K K 4 ! @F ?KJO@ )1 @F ?JN@= K@ STU A+B8MNSTUK=>ADSVU @]@= #D

f(x) = aT x + α [J:KCI?[AfJ:K =yKY5

f(x) = xT Hx [J8KC ?3AJ8K h`7.=*MO@F=`?[A3h`7]5 =]t:5 C H;>=?3@BA3CD5 Cm=:@B@DH5Em5F;<A M*H K(Af5GvqJ:EcA[?[A[t:5

Wf(x) = ec(x) J c

5FEI?l7KY5 [J:KCI?[AfJ:K CDJ8KLt85m9 5

f(x) =‖ x ‖ J ‖ · ‖ 5DE?7KL5KYJ8@B;s5 t:5 C ?QJ8@BA5m^a^f56h`7]5m^[CFJ:Kh`7]5f(x) = af1(x) J f1

5DE?GCFJ:KLt85B9 558?a ≥ 0

f(x) = f1(x) + f2(x) J f1

58?f2

EFJ8KL?!M*587:9 [J8KC ?3AJ8K(ECDJ8KLt85m9 5FE

f(x) = maxf1(x), f2(x) J f15:?

f2EmJ:KY?GM*5:7:9[J:KCI?[AfJ:K(ECFJ:KLt85B9 5DE

,P.-UKMLB8MOSVU )1 ,STF QWMNUK@S#H UK@ SVUKAB%MNSTU A+STU @]@ 5GM`J8;b=:AaKY5!MN\ 7KY5 [J8KC ?3AJ8K>CDJ8KLt85m9(5 f KYJ?QHdom(f) 5FEI? ^m\ 5mK(EF5m;kn:^56M`5FE x

?Q5F^aE6h*7*5f(x) < +∞

i C:S ?KC:MNPB8P 1[2 J:A[?fCDJ8KLt85m9(5 x0 ∈ dom(f)

58?d7KY5gMOAa@D5 CI?[AfJ:K M*5

Rn ?Q5F^_^5gh`7]5x0 + d ∈ dom(f)

= [J:KCI?[AfJ:K

q(t) =1

t[f(x0 + td)− f(x0)]

5FEI? ;sJ8KYJ`?QJ:KY5 Cc@DJ8AoEcEm=:KL?Q5pE7O@(0, 1)

wyxz|~L]~| & QRY~A

0 ≤ h ≤ k ≤ 1S/1<S<S36Y6`^KSQR027T3E9+L8I20SYEA=Y6Q107=KSCk3E9+7=9hRWX90^A

[x0, x0 + kd]

f(x0 + hd) ≤ λf(x0) + (1− λ)f(x0 + kd)

/N19@?h = (1− λ)k

Z|\0&90L8I@L8KSYEAM/4Yc7:IUWX90^A;`RK29q(h) ≤ q(k)

Z Q1WXWX9m?JQR027:I@`^KS902?9U7nQR0&/X`^KS9

(0A=QRK8Ae<>QRYE0^ApLS93H- YE0^ADICDYE9UKSCC=9U36/4A=YE_LS9 dom(f)lf9@7FAa?Q10^A=Y60gKS99JAe3c/ L8IUC=Y6N1IU9rLSYECD9U?JA=Y6Q10S0S9U3E3699Ju8Y67:A=9RZSdM9m<23EK27

f9U7:A;36Qg?U/4369WX90^A;,yY6<27D?[wSYEA/(Y69020S91l8Y Z 9RZ

|f(y1)− f(y2)| ≤ L ‖ y1 − y2 ‖,Q+,

y1ly27:QR0RA;`^KS9U36?Q102`^KS9@7TL2/4027nKS0rNRQ1Yc7:Y602/4hR9]LSK&<fQ1Y60^A

x09A

L > 0L8I<f902LrL89

x0Z

& Yf9@7FAM?Q10gN19u89\9AkLSY~ICD90^A=Yc/4j23E9m9U0

x0lSQR0&/q<fQ1KSC

z ∈ dom(f)

f(z) ≥ f(x0) +∇f(x0)T (z − x0).

& Yf9U7:A]?Q10gN19u89]9A]L89UK8u&Q1Yc7ML8YEfIUC=9U0^A=Yc/4jS3691l>3E9 ;9U7D7=YE9U0 ∇2f(x0)

9U7:AMKS0S9+W/4A=CDY6?9z7:9UWXY~b L8I >0SYE9<>Q^7:YEA=Y6N19RZ PS C F @ 1 SVU KMLB8MOSVU ! SV? B%MNF Q JNMLB%P ?STH C H UK@ STU A+B8MNSTU A+STU @]@ KM PDC:@U B8MOQJN@ KbvqJ8AaKL?

x∗ ∈ Rn ;<AaKAa;yAaEF5

fCDJ8KLt85m9(56MOA)(HF@F5FKL?3Af=n:^5EA 58?'EF587O^f5F;b5FKL?qEcA ∇f(x∗) = 0.

|\j27=9CDN198(T`^KS9kLS/10273E9;?U/17P?Q10gN19ug9Rl36/_?JQR02L8YEA=Y6Q10q0SIU?9U7D7=/1YECD9nL8KX<SC=9UWXYE9UCPQ1C[L8C=9nL89UN^Y690^A(0SI@?J9@7=7D/4Y6C=9T9JA7=K<e7=/10RAD91Z>|\0&C=9A=CDQ1KSNR9MYc?JYV369_/4YEA;`RK29\ADQ1K8AkWXY60SYEWKSW36Q8?/13yLV- K20S9]Q102?JA=Y6Q10r?JQ10gNR9Ju89\9@7FAkKS0&WXYE02YEWKSWh136Q1j>/43 ! <2C=QR<SC=Y6IJADIm`RK2YV0y- 9u8YEhR9]<>/17n3c/qLSY~ICD90^A=Yc/4j2YE36Y~ADI%$JZ

q/. T&"$pk $&$pk*'Gk*,P.-UKMLB8MOSVU )1 ,MLC:@AB%MNSTU @ K@=8AD@U B8@ Kkt85 C ?Q5:7@

dM*5

Rn 5FEI?q7KY5pMOAa@F58CI?[AfJ:KsM*5pM*5DEDCD5mKL?Q5vqJ*7@^f=[J8KC ?3AJ8K

f=`7 vqJ8AaKL?

xEcA

f ′(x; d) = ∇f(x)T d < 0.

Page 81: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W 4 4 4 K K "

xa

f(a)

f(x)

(D)

Z 2Q102?JA=Y6Q10?JQ10gNR9Ju89L8YEICD90^A=Yc/4jS3691ZV,*/a?JQRKSCDj>9XL89 f9U7:A\ADQ1K FQRKSC[7\/1KL89@7=7=K27_L89q36/aAD/10Sh19U0RAD9

(D)

& QRY~Ad9U7:AMKS029+L8YECD9U?JA=Y6Q10L89zLS9U7D?J90^AD9m9JA

tKS0CDI93!<>Q^7:YEA=YEFZ|\07=KS<S<fQR7=9

t7=K<e7D/4WXWX90^A;<>9A=YEA\<fQ1KSC`^KS9_3 - /1<S<SCDQu8YEW/OADYEQR0pL8Kr<SCD9WXY69CkQRCDL8CD9_7:KSY6NO/40^A=9m7=Q1YEAkN/136/1jS369

f(x + td) ≈ f(x) + t∇f(x)T d.

d\Q10>?_Q10r/f(x + td)− f(x) = t∇f(x)T d < 0

l8Y Z 9RZf(x + td) < f(x)

Z,!9U7pWqIA=wSQ8L89@7pLS9L89@7=?90^A=97=Q10^ArL89U7aWqIA=wSQ8L89@7aYEA=IUCD/4A=Y6N19@7e<fQ1KSC&36/ WqY60SY6WXY67D/OADYEQR0 L89U7aQ10>?BA=Y6Q10>7LSY~ICD90^A=Yc/4j23E9@7q7=KSCRn LS/10273E9@7=`^KS9U3E369U7qKS0S9rL8Y6C=9@?BADYEQR0 L89rL89U7D?J9U0^A=9&9U7:AX?[wSQRY67=Y69po?[w2/1`^KS9pY~ADIC[/OA=Y6Q10so<>/4C=A=Y6CzL89U7+YE08QRC=W/4A=Y6Q1027_hRI0SIUCD/13E9UWq9U0^A+3EQ8?/13E9@7Z*|\0WXY60SYEWXYc7:9p/436Q1C[7_36/&QR02?BADYEQR0 LS/4027z?J9JA=A=9aLSYECD9U?JA=Y6Q10

! WXYE0SY6WXY67D/OADYEQR0 KS02Y6L8Y6CD9U?BADYEQR0S0S9U3E369Q1K CD9U?[w29C[?[wS936YE0SI@/4Y6C=9 $BZk,y97D?[wSIW/ hRI0SIUCD/13\9@7FAL8QR0S0SI?JYEb LS9U7D7:QRK27! 3H- Y602L8Yc?J9mCD9<SCDIU7=90^A=9]Yc?JYy3 - YEA=IUCD/4A=Y6Q10V$

U MNB8MNQ JNMN=%QB%MNSTUS1 w2Q1Yc7:Y6Cx0Z

B8PC:QB8MOSVUk1 wSQRY67=YECnKS029+L8YECD9U?JA=Y6Q10rL89+L89@7=?90^A=9

dkZ & - YE3!0y- 9U0r9Ju8Y67:A=9_<2/R7lS/1C=C JAkL89m3 - /13EhRQ1CDY~ADwSWX91Z

& Y60SQ10xk+1 = xk + tkdk,/N19@?

tk > 0AD93!`^KS9

f(xk+1) ≤ f(xk + tdk)l ∀t ∈ (0, δ)

ZQ1K2Ck36/7=KSYEA=91l8QR0r<>Q^7:9UCD/

gk = ∇f(xk)Z

i C:SV?ST=8MNB8MNSTU 1[2W4u=:K(Ep^o= @D5 C Y5F@C Y5l^_AaKYH=:Aa@F5 EA tk;yA_K(A_;yAoEm5u5B9(=`CI?Q5F;s5mKL?q^o= [J8K)CI?3AJ8K

θ(t) = f(xk +tdk) ^f5KLJ*7t:5D=`7gd.@F=*MOA5mKL? gk+1

=76vqJ8AaKL?xk+1 = xk + tkdk

5FEI?SJ8@ ?! YJ8d`J8KY=:^ ^o=sMOAa@F58CI?[AfJ:Kdk

wyxz|~L*~L| & YtkWXY60SYEWXYc7:9z36/qQR02?BADYEQR0

θ(t)l/436Q1C[7

θ′(tk) = 0Zf|\C@lS369U7kCDihR3E9@7kL89z?/136?KS3!L893c/XL8ICDY6N1I9

θ′ ! ?[w2/:Ec02/1h19 $.L8QR0S0S9U0RAθ′(t) = ∇f(xk + tdk)T dk.

.0&<>/4C=A=Yc?JKS36YE9UCkQ10r/θ′(0) = gT

k dk

θ′(tk) = gTk+1dk.

Page 82: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

% > 4 K \ K K 4 ! ,!/ KMLC:@AB8MOSVU K@ JNQ ?KJOH = VC Q U K@ ?@U B8@r9@7FA_369qNR9U?JA=9K2C

d02Q1CDWqIX`^KSYWqY60SY6WXY67=9

f ′(x; d)Z - 9@7FAL8QR02?_36/XL8Y6CD9U?BADYEQR0pQR<S<fQR7=I9m/4K&h1C[/1L8Y690^A

dk =−gk

‖ gk ‖.

(0<SC[/OADY6`^KS9Rl8Q10&K8A=Y636Y67=9+/4K27D7=YV36/qN19UCD7=Y6Q10a02Q10&0SQ1CDWXI91l ; 2 dk = −gk.

,!/F PB S @ KH C:Q KMN@U B ! Q1K+L89.<S3EK>7yh1C[/40>L89P<>9U0^A=9%$!?[wSQ1Yc7=Y~A!3c/kL8Y6C=9@?BADYEQR0+L89(<S36K27yhRCD/102L89(L89@7=?90^A=9o?[w2/R`RK29\YEA=IUCD/4A=Y6Q10yZ>|\0&C=9UW/4C[`RK29_`^KS9RlSLS/40>7k?J9+?/R7lg369U7;LSYECD9U?JA=Y6Q1027k7=K2??9U7D7:Y6N19@7n7:QR0RAnQ1C=A=w2Q1h1QR02/4369U7UZ 39U7:A_?36/1YEC_`^KS9q36/&L8Y6C=9@?BA=Y6Q10L89<23EK27]hRCD/102L89z<f90^AD9<f9K8A- A=CD9/4CDjSYEA=C[/4Y6C=9UWX90^A\W/4K2N/1Y67=9z9U0?9q`^KSY?JQR02?J9UC=029.3c/\L8Y6CD9U?BADYEQR0L8KWXYE0SY6WzK2W&Z4,!9T?U/17L89@7yQR02?BADYEQR027`RK>/1L8C[/OADY6`^KS9@7!9@7FA<2/4C=A=Yc?JK23EY6iCD9WX90^A#YE027:A=CDK2?JA=YE9JA]Q10NR9CDCD/XLS/1027;3c/e7=9U?BADYEQR07:K2YENO/40^AD9z?JQRWqWX9U0RA]/4WXI36YEQRC=9UCMoX<f9KL89zCD/1Y67M?J9JA=A=9qL8YECD9U?JA=Y6Q10<fQ1KSC]7:K2YENgCD93c/XN/13E36I9_9A;0SQ10&36/q<f90^A=9mNR9C[7n36/X7=Q136K8A=Y6Q10!Z

q )B k'.*)J"+$p *"+T$&)B k'(#)J"+&'T"+y%1+%&kkd]/4027T369]?U/17nLV- KS0S9\Q102?JA=Y6Q10p`^K2/RL8C[/OA=Yc`^KS9;QRC:AD9WX90^Ak?JQR0^NR9Ju89 ! L8Q10>?\/R7=7=Q8?JY6I9\oKS0S9]W/OADC=Yc?J9_L8I >0SYE9<fQR7=Y~ADYENR9%$BlOQR0<f9KSA(WqQR0^A=CD9C ! ?JFZ1,yKS9U0gj>9UC=hR9C 5 9 $`^KS9n3c/MWXIJADwSQ8L89;L8Kqh1C[/1L8Y690^AP/N19@?.CD9U?[wS9UCD?[w29U7*3EY60SIU/1YECD9U79JuS/R?BA=9@7n<SCDIU7=90^A=9_KS0S9+?Q10gN19UC=hR902?9M36YE02IU/4Y6CD9]/NR9U?]KS0pAD/4KSupIh^/43

(

r − 1

r + 1

)2

Q+,r9@7FAM369+C[/4<S<fQ1C=A;90^ADC=9+3c/X<S36K27\h1C[/402L89_9A\3c/X<S3EK>7;<f9JA=YEA=9NO/4369KSC\<SC=QR<SCD9+L89+3c/XWX/4A=CDY6?9z/17D7=Qg?YEIU9+oX36/Q1CDWX9q`^K2/RL8C[/OA=Yc`^KS91Zf(363E9XL89UN^Y690^Am7:QRK27:b 36YE02IU/4Y6CD9z`^K2/402L36/eQ102?JA=Y6Q109U7:A]W/43P?Q102L8YEA=Y6Q1020SI9Rlf?1- 9@7FA=b o4b L8Y6CD9`^K2/40>L

rL89NgY690^A;h1C[/40>LVZ

! p p" " " ` S& QRY~A@l2LS/40>7

Rn lq(x) =

1

2xT Ax,

/N19@?A7:5gWXIJADC=Yc`^KS9XL8I 202YE9q<>Q^7:YEA=Y6N19RZ(0KS0<fQ1Y60RA

x ∈ Rn LSY~ICD90^AmL893 - Q1CDYEhRYE0S9 ! 3 - K20SY6`^KS9qWXYE02YEWKSWh136Q1j>/43yL89q$BZ>,y9mh1C[/1LSYE9U0RAkL89

q9U7:A ∇q(x) = Ax

Z|\0r7:9m<S3c/1?9_90&KS0r<>QRYE0^Ax0Z>|\0&/4YEA;KS0S9m/4<2<SC=Qu8Y6W/OA=Y6Q10&LV- Q1C[L8C=9mqL89

q90

x0Zf|\0pQRj8A=Y690^A

q(x) =1

2xT

0 Ax0 + (x− x0)T Ax0 +

1

2(x − x0)

T A(x− x0).

|\0 ?[w29C[?[wS9W/4Y60^A=9U02/40^AUlkov<2/4C=A=Y6CL89x0lk369<fQ1Y60^A

x∗ A=9U3_`^KS9 ∇q(x∗) = 0lk?J9`^KSYmL8Q1020S9

x∗ =x0 + dN

l8QD,dN = −A−1∇q(x0)

9@7FAn3c/zL8Y6C=9@?BADYEQR0pL89;9nA=Q10!ZS|\0pQ1j27=9CDN19\`RK29]?9JA:AD9]LSYECD9U?JA=Y6Q10eQRKSCD0SY~AY6027FA[/40^AD/10SIWX9U0RA ! 7=/1027C=9@?[wS9C[?[wS9T3EY60SI@/4Y6C=9 $!369nWXYE0SY6WzK2W L89n3c/\Q102?JA=Y6Q10?/1Cx0 +dN = x∗ = 0 ! ?JFZ 2hRKSC=9 Z $JZS,(- /436h1QRC=YEA=wSWX9m?JQR0gN19CDh19]L8Q10>?\9U0rKS0S9+7=9KS369_Y~ADIC[/OA=Y6Q10!Z

Q1KSCkK20S9_Q102?JA=Y6Q10fL89UK8uaQRY67kLSY~ICD90^A=Yc/4j23E9RlSQ10&IU?C=YEAk3H- /4<S<2C=Qu8Y6WX/4A=Y6Q10&LV- QRCDLSC=9m

f(x) = f(xk) +∇f(xk)T (x− xk) +1

2(x− xk)T∇2f(xk)(x − xk).

|\0?[wS9C[?[wS9]9027=KSYEA=9xk+1

AD93!`^KS9 ∇f(xk+1) = 0Z>|\0aADC=QRKSN19

xk+1 = xk + dN , dN = −[

∇2f(xk)]−1∇f(xk).

Page 83: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

4 K 4 4 K

x∗

x

dN

−∇f(x)

Z d\Y6C=9@?BA=Y6Q10>7kL89 ;9nA=Q10&9JAMLSK&hRCD/RL8Y690^A

QRWXWq9\<>QRKSCn3E9@7(QR02?BADYEQR027n`^K2/1L8C[/OADY6`^KS9@7lR36/L8Y6C=9@?BA=Y6Q10p9JAn3E9]<2/R7GL89_L8IU<S36/R?J9UWq9U0^AT7=Q10^A Su8I@7Z & Y3 - QRCDL8CD9LS9?JQR0gN19CDh19U02?J9L89UNgYE9U0RAr`^K2/1L8C[/OADY6`^KS9Rl(3 - /1<S<S36Y6?U/OA=Y6Q10 L89?9JA:AD9YcL8I9LS/4027p369?/R7ehRI0SIUCD/13;9U7:Ap36YEWXYEA=IU9<>/4C_3E9e?Q1t8A+L8K?/136?KS3(L89X3 - Y60gN19C[7=9qL8K M9U7D7:Y6909JAm<2/1Cm3E9@7m7:IUN1iCD9U7\wg5g<>Q1A=wSi@7:9@7m`RK2YPhR/4C[/40^ADY67D7:9U0^A]?J9A:AD9?Q10gN19UC=hR902?91Z(0 <SC[/OADY6`^KS9RlV36/LSYECD9U?JA=Y6Q109U7:Az/4<S<2C=Q8?[wSIU9qYEA=IUCD/4A=Y6N19UWq9U0^A ! ?BFZ#?JQ1K2CD7+4iWX9e/40S02I9 $]9A02Q1K27k/13E36Q1027nY6363EK27:A=CD9C;?9JA=A=9_<>Q^7=7=Y6jSYE36YEA=I+/N19@?M3c/qWXIJA=w2QgLS9+L8Krh1C[/1L8Y690^A;?JQR0 FKSh1K2I1ZGp p " "+ &"Y` S+ L ! )"+,!9U7eWXIA=wSQ8L89@7aL89@7aL8Y6C=9@?BADYEQR027a?Q10 FKShRKSI9@7e7=Q10^ApL89@7eWXIJADwSQ8L89U7aY~ADIC[/OA=Y6N19@7e`^KSY lG/4<2<S3EYc`^KSI9@7eo KS029QR02?BADYEQR0&`^K2/RL8C[/OA=Yc`^KS9m?JQR0^NR9Ju89_o

nNO/4CDY6/1jS3E9@7lS?Q102LSKSY67=90^A;oq3H- Q1<SA=Y6WzKSW 90

nIAD/4<f9U7k/1Kr<S3EK>7Z

,P]-KUKMLB%MNSTU 1[2AH:?P=:KL?7KY5 ;>=?3@BA3CD5uE `;sH:?3@BA3h`7]5yM*H K(Af5vNJEAP?3APt85 M*5:7:9 t858CI?Q5:7@mEpKYJ:K KL7^aE

x5:?

yM`5Rn EFJ8KL?GMOA[?PE & # Rpv)=@l@m=IvvNJ:@I? A

EcAxT Ay = 0

i C:SV?ST=8MNB8MNSTU 1 A

pt:5 CI?587@mE<KYJ8K%KL7^aEkM`5

Rn p ≤ n EmJ:KY?;s7(?7]5F^_^5m;s5mKL?<CFJ:K 7:dY7]HFE A_^oE>EmJ:KL?^aAaKYHD=A_@D5m;s5mKL?qAaKM*Hvq5mK)M]=:KL?[E wyxz|~L*~L| & QRY~A

αiA=9U367M`RK29

p∑

i=1

αidi = 0. ! Z $

|\0&WzK23~ADYE<S36Y69 ! Z $noqh^/4K2?[wS9]<2/1C AlSQR0pQRj8A=Y690^A

p∑

i=1

αiAdi = 0. ! Z $

.0 WzKS3EA=Y6<S36Y6/10RA ! Z $q<>/4C dT1

l.Q10A=CDQ1K2N19α1d

T1 Ad1 = 0

l(<2KSY67D`^KS9369U7NR9U?JA=9K2CD7e7=Q10^AeWK8A=KS9U3E369WX9U0RA?Q10 FK2h1KSI@7Z>|\0rL8IULSKSY~AM`^KS9α1 = 0

ZS(0rC=IU<>IAD/10RAk369_<SC=Q8?9U7D7:K27n/NR9U?d2, . . . , dp

lSQR0&/1C=CDY6N19\oα1 = α2 =

· · · = αp = 0ZSdMQR02?]3E9@7kN19@?BA=9UKSC[7

d1, . . . , dp7:QR0RAn36YE0SI@/4Y6C=9UWX90^AkYE02LSI<f902LS/10^AD7UZ

& QRY~A;3c/Q102?JA=Y6Q10`^K2/1LSCD/4A=Yc`RK29]?Q10gN19ug9

q(x) =1

2xT Ax + bT x + c.

Page 84: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

% > 4 K \ K K 4 ! i C:MNUKA+MN?@ @ JOQ F PB S K@ (|\0<>/4C=A_LV- KS0<>QRYE0^A

x0lVQ10WqY60SY6WXY67=9

q7:K>??J9@7=7=Y6N19WX9U0RA_7:K2YENO/40^A

nL8Y6C=9@?BADYEQR027d0, d1, . . . , dn−1

?Q10 FK2h1KSIU9U7\<2/4C\CD/1<S<>QRC:A]o36/aW/OA=CDYc?J9AlfY Z 91Z

dTi Adj = 0

l2<fQ1KSCi 6= j

Z,!/7=IU`^KS90>?J9 xk9@7FATL8I 202YE9;<>/4C

xk+1 = xk + tkdkl^Q+,

tkWXYE02YEWXYc7:9

θ(t) = q(xk + tdk)Z^,!9;<>QRYE0^A.QRj8A=9U0gKoX3c/

nb iWX9_YEA=IC[/OADYEQR0yl8Y Z 9RZ

xn = xn−1 + tn−1dn−1 = x0 +

n−1∑

i=0

tjdj

9U7:A;3 - QR<8A=Y6WzKSW L8Kr<SCDQ1jS36iWX9 ! ∇q(xn) = Axn + b = 0$JZ

PS C F @ 1 5FEld.@F=*MOA5mKL?[ElEI7`CICF5FEBEcA EuCF=:^[CI7^HFE=`7:9bvNJ:A_KL?PExk

dHmKYHF@FHDESv)=:@^o=<;sH8?% YJM*5 M*5DE6MOAa@D5 C ?3AJ8K EuCFJ:K 7:d 7*H5FElEm=`?[AaE [J8KL?

gTk+1di = 0, ∀i ≤ k.

wyxz|~L]~| P/4C]wg5g<>Q1A=wSi@7:9q7:KSC_369?[wSQRY~uQR<8A=Y6W/43PL89tkl>QR0/

gTk+1dk = 0

Z QRWXWq9q9@7FA`^K2/1LSCD/4A=Yc`RK291l8Q10r/q3E9@7kCD93c/OA=Y6Q10>7

gk+1 = gk + tkAdk,

gk+1 = gi+1 +

k∑

j=i+1

tjAdj , i < k.

d- QD,&369mCDIU7=KS3~A[/OA@Z

S C:STJNJOQWMLC:@ )132xk+1 Cm=:^[CI7O^fH <^F\ A[?Hm@F=`?3AJ8K k

M`5^o=>;bH:?! YJM*5<M*5FE MOA_@D5 C ?3AJ8K(E CFJ8K 7:d 7]HI5FE ;yA_K(A_;yAoEm5 qEI7@l^f5EFJ*7.E Q5FEQv)=*CD5p=<KL5Vk = x0+ lind0, . . . , dk

wyxz|~L]~| #x 3H- Y~ADIC[/OADYEQR0

k + 1lV3E9XhRCD/RL8YE9U0^A

gk+19U7:A_QRC:ADwSQ1hRQ102/13#o

d0, . . . , dklyL8Q10>?QRC:bA=w2Q1h1QR02/43yo

VkZ - 9U7:Ak36/?Q102L8YEA=Y6Q10rLV- Q1<8ADYEW/13EYEA=I+L8Kr<SCDQ1j23EiUWq9_L89mWXY60SYEWXYc7=/4A=Y6Q10L89

q7:KSC

VkZ

S C:STJNJOQWMLC:@ )1 =<;sH8?% YJM*5 M*5DEuMOAa@F58CI?3AJ8K ECFJ8K 7:d 7]HI5FE =v(vL^_A[h`7]H5k7KL5[J8KC ?3AJ8K h`7.=*M.@m=?3A[h*7*5[J:@ ?Q5F;s5mKL?CFJ:KYt:5B9 56M*5

Rn CFJ:KLt85m@cd`55mK =`76vL^P7OE nA[?Hm@F=`?3AJ8K E

wyxz|~L]~| & Q1YEAgk = 0

<fQ1KSCk < n

lS7=Q1YEAgn = 0

?/1CVn = Rn LV- /4<SCDiU7T369]?Q1CDQ13636/1YECD9 Z#"1Z

;;029mWX/10SY6iCD9_7=Y6Wq<23E9mL89+?Q1027:A=CDKSY6C=9]3E9@7;L8Y6C=9@?BADYEQR027n?JQ10 FKSh1KSIU9U7nYEA=IUCD/4A=Y6N19WX9U0RAk9@7FA"RZ

d0 = −g0SZ & Ygk+1 6= 0

lS/13EQRCD7

βk =gT

k+1Adk

dTk Adk

,

dk+1 = −gk+1 + βkdk.

,y9\?JQg9 <e?JY690^Aβk9@7FA.?U/43c?JKS36I;L89M7=Q1C=A=9k`^KS9

dTk Adk+1 = 0

Z1,!/_WXIA=wSQ8L89;7U- /1<S<f9363E9;/13EQRCD73c/-F PB S K@ KHVC Q MO@U BIA+STU%H VHKPfZ!(36369e<>9UK8A A=CD9eK8ADYE36Y67=I9a<>QRKSC+WXYE02YEWXYc7:9UCzKS0S9eQ102?JA=Y6Q10 `^K2/1LSCD/4A=Yc`RK29XQ1C=A=9WX9U0RA?JQR0gN19Ju89Rl?1- 9@7FA=b o4b LSYECD9a<fQ1KSCCDIU7=Q1K>L8C=9pKS07:587:A=iWX9LV- IU`^K2/4A=Y6Q1027q36YE0SI@/4Y6C=9@7XL8Q10^A36/W/OADC=Yc?J9r9U7:AeL8I >0SYE9<fQR7=Y~ADYENR91Z

& QRY~AMoqCDIU7=Q1K2L8CD9]3E9m7=5g7:A=iUWX9Ax = b ! Z $/N19@?

AL8I20SY69<fQR7=Y~ADYENR91ZT|\0 <f9K8A&C=I@7:QRK2L8CD9?J97=587FADiWX9oA=C[/N19UCD7X3c/WXYE02YEWXYc7=/4A=Y6Q10 L893c/QR02?BADYEQR0`^K2/1LSCD/4A=Yc`RK29\QRC:AD9WX90^A;?JQR0gN19Ju89

q(x) =1

2xT Ax− bT x

Page 85: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

4 K 4 4 K ^LSQ10^Ak3E9mhRCD/RL8YE9U0^AT9@7FA ∇q(x) = Ax− b

ZSd\9+7=Q1C=A=9m`^KS9m7:Y ∇q(x∗) = 0/436Q1C[7

x∗ 9@7FAM7=Q136K8A=Y6Q10rL89 ! Z $BZ" J TS C:MNB F @ KH VC Q MO@U BGADSVU %H TH P)?SVH C JN@= SVUKAB%MNSTU =3 HKQ C QWB8M HK@=U MNB8MNQ JNMN=%QB%MNSTU

k = 0. x0L8QR0S0SIRl

g0 = ∇q(x0) = Ax0 − bld0 = −g0

B8PC:QB8MOSVUk ≥ 0. xk

lgk9A

dk?JQR0S0gK27

/136?KS3 ! 9Ju8<S36Y6?Y~AD9%$nL8Kr<2/R7kL89+L8I<236/R?J9WX9U0RA

tk = − gTk dk

dTk Adk

Y67=9+o FQ1KSCxk+1 = xk + tkdk

MQ1KSNR9363E9mL8Y6CD9U?BADYEQR0&LSK&hRCD/RL8Y690^A;?JQ10 FKSh1KSI

βk =gT

k+1Adk

dTk Adk

dk+1 = −gk+1 + βkdk.

(0r<SC[/OADY6`^KS9_QR0&/1C=C JAD9\369U7nYEA=IUCD/4A=Y6Q1027nL8i@7k`RK29 ‖ gk ‖9U7:A 7=K<e7=/1WXWq9U0^A\<f9JADY~A@Z 9JA=A=9mWXIJADwSQ8L89_9U7:AADC=i@7;<fQ1<SK236/1YECD9m9JA]<>9UK8A]?Q102?KSC=CD90>?J9Ck3c/WXIJADwSQ8L89LS9 wSQ1369U7g5 !q w>/4<yZ $k`^K2/102Lr36/aL8YEWX9U027:Y6Q10LS93 - 9U7=<2/1?9

n9U7:AkA=CDiU7nIU3E9UN1I9RZ

,.- /13EhRQ1CDY~ADwSWX9r?Y~b L89@7=7=K27q<>9UK8AaIUhR/4369WX9U0RAG A=CD9r/RLS/4<SA=I/4K8u Q102?JA=Y6Q1027X02Q10`RK>/1L8C[/OADY6`^KS9@7L8Y~ICD90SbADY6/1jS369U7UZy|\0INgYEA=9/436Q1C[7M369X?/43c?JK23LSK M9U7D7:Y6909JAm3E9@7]<>9UC:QRC=W/40>?J9U7]7=Q10^A\ADQ1K FQRKSC[7\7=KS<fICDYE9UKSC=9@7_op?9363E9@7LS9m36/qWXIJADwSQ8L89+L8Krh1C[/1L8Y690^A@Z" J TS C:MNB F @= KH C:Q KMN@U BGA+STU%H VHKP K@JN@B8A @C @@ @=X@ByiSVJOQ 6M M C:@U MNB8MNQ JNMN=%QB%MNSTU

k = 0. x0L8QR0S0SIRl

g0 = ∇q(x0) = Ax0 − bld0 = −g0

B8PC:QB8MOSVUk ≥ 0. xk

lgk9A

dk?JQR0S0gK27

/136?KS3!L8Kr<2/R7kL89mL8I<236/R?J9WX9U0RA;<>/4CnC=9@?[wS9C[?[wS9]3EY60SI@/4Y6C=9tk = arg min

t>0f(xk + tdk)

Y67=9+o FQ1KSCxk+1 = xk + tkdk

MQ1KSNR9363E9mL8Y6CD9U?BADYEQR0&LSK&hRCD/RL8Y690^A;?JQ10 FKSh1KSI

βk =‖ gk+1 ‖2‖ gk ‖2

, ! *369JA[?[wS9C=b 99UN19U7 $

βk =gT

k+1(gk+1 − gk)

‖ gk ‖2, ! #QR36/^b Y6jSY6iCD9%$

dk+1 = −gk+1 + βkdk.

6@F QC" H @ )132 v@FeDEnAP?QHF@m=?3AJ8K(E J8K =pd`HFKYHm@DH Rn % ^3=`7(?M`J8KCN@FHFA_KAP?3Af=:^aAoEm5F@^F\ =^ dJ8@BA[?% ;s5S5FKvqJ:Em=:KL?

dk = −gk =7<^aA587kM*5FE [J:@c; 7^5FEM(7<d.@F=*M.Af5FKY?!CFJ:K 7:dY7]H ?QJ`7?Q5DEl^5FE nA[?Hm@F=`?3AJ8K E

Page 86: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

+ % > 4 K \ K K 4 ! q[ a#) )J!m*)J"+ !"+%& 'T"+! #z)J*k

;;0r<SCDQ1j23EiUWq9mLV- Q1<SA=Y6WqYc7D/OA=Y6Q10rLS/1027Rn <2C=9U02La3c/qQ1CDWX9_h1I02IC[/4369]7=KSY6N/10^A=9

Y60SYEWXYc7:9UCf(x)7:QRK27gi(x) ≤ 0, i = 1, . . . , p

hj(x) = 0, j = 1, . . . , q

x ∈ SQ+,flgilhj7:QR0^AkLS9U7nQR02?BADYEQR027kL89

Rn LS/4027 R9JA

SKS0r9027=9WjS3E9+LS9

Rn Z|\0z/\C=9U<SCDIU7=90^A=I(Yc?JY^ADC=QRY67!AF5g<>9@7*L89n?JQR0^A=C[/4Y60RAD9U7 ?Q10^A=C[/4Y60^A=9@7!L8I@?JCDY~AD9U7*<2/4CL89U7#Y60SIh^/436Y~ADIU7UlU?Q10^A=C[/4Y60^A=9U7L8I@?JCDY~AD9U7G<2/4CnL89U7GIh^/436Y~ADIU7.9JAk?JQ10^ADCD/1YE0^A=9@7(L8I@?JCDY~AD9U7G<2/4CTKS0e9U027:9UWzjS369 ! 0SQ10a0SIU?9U7D7=/1YECD9WX90^A.9u8<S3EYc?JYEA=9 $BZ8,y9<SCDQ1jS36iWX9k?JQR027=Y67:A=9nL8Q10>?ToMA=CDQ1K2N19C<2/1C=WXY^A=QRK27369U7x`^KSYS7=/4A=Yc7FQR0^A?Q10 FQRYE0^A=9UWX90^AA=Q1KSA=9U7369U7?Q10^A=C[/4Y60^A=9U7KS0

x`^KSYPC=9U02L

fWqY60SY6WzKSWrZ#|\09C[/p?Q1WXWX9q<2C=I@?JIULS9WXWX90^A]3c/rL8Yc7FADYE02?JA=Y6Q1090^A=CD9qWqY60SY6WzKSW 3EQ8?/13P9JAWXYE02YEWKSW hR3EQRj2/43 Z

![ l S!`& !`"+ `+|\0r?JQ10>7:YcL8iCD9_3E9_<SCDQ1j23EiUWq9

(PE)

YE0SY6WXY67=9Cf(x)7=Q1K>7hj(x) = 0, ∀j = 1, . . . , q.|\07:K2<S<>Q^7:9UCD/369U7TQ10>?BA=Y6Q10>7TFl

h1l6ZZZUl

hq?JQR0RADYE0gKSWX9U0RA;L8YEICD90^A=Yc/4jS369U7UZ

PS C F @ 1f/ Ax∗ 5DE?S;<AaKAa;s7; ^JCm=:^LvNJ`7@^5lvL@DJ`n^feF;b5

(PE)5:?!EFJ*7OEp^m\ OvqJ*?% YeFEF5 h`7]5^5FEd.@F=

MOA5mKL?[E ∇hj(x∗) j = 1, . . . , q EFJ8KL?l^aA_KLHD=:Aa@D5m;s5mKL?GA_KM`Hvq5mKM]=KL?[E =:^J8@mEA_^G5m9`AaEI?Q5 q

;s7^P?3A vL^aA3CF=`?Q5:7@mEyM*5 =:d.@F=:K`d5

λj 1 ≤ j ≤ q ?5m^oEuh`7]5

∇f(x∗) +

q∑

j=1

λj∇hj(x∗) = 0. ! Z $

wyxz|~L]~| y,!/pL8IWXQR027FADCD/4A=Y6Q109@7FA]j2/17=I9z7=KSC]3E9zA=wSIUQ1CDiWX9LS9z3c/Q10>?BA=Y6Q10YEWX<S36Y6?Y~AD99A\QR00S9L8QR0S0S9UCD/Yc?JY;`^Ky- KS0S9L89@7=?C=Y6<8A=Y6Q107=Q1WXW/4Y6C=9RZ,.- 9U027=9Wzj23E9L89@7X<>QRYE0^A[7eL8YEAD7CDIU/13EYc7=/1jS369U7Ul#?1- 9@7FA=b o4b L8Y6CD9`^KSYG7D/OADY67:Q10^Az7=YEWKS3EAD/402IWX90^AmA=QRK8A=9@7m369U7+?Q10^A=C[/4Y60^A=9U7UlV9U7:A+KS0S9p7:KSC=/1?9SLS9

Rn Z!,(- wg5g<>Q1A=wSi@7:9e7=KSC+369U7h1C[/1LSYE9U0RA ∇hj ! wg5g<fQ4A=w2iU7=9+L89_C=IUh1KS3c/4CDY~ADI%$GY6Wq<23EYc`^KS9+`^KS9! Y $&,!/X7:K2C:/R?J9 S

9@7FA;36Q8?/13E9UWq9U0^AnK20S9_N/1C=Y6IJADImL8Y~ICD90^ADY6/1jS3E9+LS9_LSYEWX90>7:Y6Q10n− q

Z! YEY $&,y9]<S36/10AD/10Sh19U0^ATo S

90x∗ 9U7:AnKS0p7=Q1K27:b 9U7=<2/1?9;/:<0S9_L89]LSYEWX90>7:Y6Q10 n− q

9An<f9KSA A=CD9]L8I20SY90Q102?JA=Y6Q10rL89U7khRCD/RL8Y690^AD7 ∇hjZ

Q1WXWX9]Q10&0S9m7- YE0^ADICD9U7D7:9]`RK!- /1K8upL8Y6C=9@?BADYEQR027noz<2/1C:ADYECkL89x∗ l8QR0&/1<S<>9U3E369C[/+<236/10aA[/40ShR90^Ano S

90x∗369+7:QRK27Fb 9U7=<2/R?J9

M = y ∈ Rn : Jy = 0Q+,

[email protected]^/1?Q1jSY690L89@7.?JQR0^A=C[/4Y60RAD9U79U0

x∗ ! ∇hj(x∗)T y = 0

l ∀j $JZD;M0S9\?Q102L8YEA=Y6Q1002IU?J9@7=7D/4Y6CD9T<fQ1K2CG`^KS9x∗ 7=Q1YEAMKS0WXY60SY6WzKSW 36Qg?U/43#L89

f7=KSC

S9@7FA_`RK!- Y63*0!- 9ugYc7:A=9+<2/R7ML89qL8Y6C=9@?BA=Y6Q10>7ML89qL89U7D?J9U0RAD9zLS/1027;369z<S3c/40AD/10Sh19U0^A

MZ((3EK>7k<SCDIU?JYc7=IWX90^A

dT∇f(x∗) ≥ 0, ∀d ∈M?J9X`^KSY#YEWX<S36Y6`^KS9 ! ?U/4C −d ∈ M$;`^KS9

dT∇f(x∗) = 0QRK90>?JQ1CD9z`^KS9 ∇f(x∗)

9U7:A\QRC:ADwSQ1hRQ102/13!oM ! ?JFZ2hRKSC=9 Z $BZ>|\CkKS0S9mCD9<2C=I@7:9U0RA[/OADYEQR0aL8K7:QRK27:b 9@7:<2/R?J9

M⊥ 9@7FA ! ?BFZ2?[w>/4<yZ " $M⊥ =

y ∈ Rn | y = JT λ

LV- Q+,r369]CDIU7=KS3EAD/4AUZ

Page 87: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

K) K K > >K

@F ?JN@ 1 J8AP? ^f5GvL@DJ`n:^em;s5 A_KAa;yAaEF5m@

f(x) = x21 + x2

2 J*7.Ex1 + x2 = 1.

K H CB@cAP? ^5FECFJ:KMOAP?3AJ8K(EMq\ Jvq?3Aa;b=:^aA[?HFE CD5 h*7OA M*J8K(KY52x1 + λ = 0

2x2 + λ = 0

x1 + x2 = 1.

K Cm=:^[CI7O^qMOA_@D5 C ?M*J8K(KY5^o=<EFJ8^P7?3AJ8Kx∗

1 = 1/2 x∗2 = 1/2 λ = −1

-

6

1

1

x∗

∇f(x∗)

O Z .36/10aA[/40ShR90^A M

9AnhRCD/RL8Y690^A

|\0XCD9W/4C[`^KS9TLSQ102?k`^KS9k369U7P?Q102L8YEA=Y6Q10>7PLV- QR<8A=Y6W/436Y~ADI ! Z $*Q1CDWq9U0^AP/NR9U?T369U7P?JQR0^A=C[/4Y60RAD9U7KS07=587FADiWX9LS9n+q

I@`RK>/OA=Y6Q10>7yon+q

Y602?JQR0S0gKS9U7UZUd]/40>7y369G?/R7Vh1IU0SIC[/43R?J9.7=587FADiWX9(9U7:A!0SQR0m3EY60SI@/4Y6C=9(9A*L8QRY~AK JA=CD9PCDIU7=Q136K<>/4C_KS0S9XWXIJADwSQ8L89XY~ADIC[/OA=Y6N19RZ(0/4YEAm?9U7]WXIJADwSQ8L89U7_YEA=IUCD/4A=Y6N19U7UlC=9Uh1CDQ1KS<fI9@7\7=Q1K>7\369X0SQ1W LS9<.C=QRh1C[/4WqbW/4A=Y6Q10 ;QR0,!YE0SI@/4Y6C=9RlV<SCDQ1h1CD9U7D7=90^A]36936Q10ShL89eL8Y6CD9U?BADYEQR027mL89eL89@7=?90^A=9 FK>7=`^Ky- o?J9`^KS9369U7+?Q102L8YEA=Y6Q10>7Ly- QR<8A=Y6W/436Y~ADI&7=Q1Y690^Aq/1<S<SCDQu8YEW/OADYENR9WX90^A7D/OADY67:/4YEA=9@7Z w2/R`^KS9aWKS3~ADYE<23EYc?/4A=9K2CXLS9&,*/4h1C[/402h19<f9K8A=b A=CD9Y60^A=9UC=<SCDIJADIm?JQ1WXWX9_KS0r<SCDY~uaW/1C=hRYE02/13VL89m36/qCD9U7D7:QRKSC[?J9 Su8I9m/4Kr7:9@?JQR02LaWX9UWzjSCD9 ! λj = −∂f/∂hj$BZ

,P]-KUKMLB%MNSTU 1[2 2 @ QVC:QWU VMO@U \ 5DE? ^f= [J8KC ?3AJ8K L M*5Rn × Rq M]=K(E

RM*H K(Af5v)=@

L(x, λ) = f(x) +

q∑

j=1

λjhj(x). ! Z $,*/+?JQR02L8YEA=Y6Q10aLy- QR<8A=Y6W/436Y~ADI\L8Ke<2C=9UWqY69CTQ1C[L8CD9k9Ju8<SCDYEWX9]L8Q102?;3E9M/4YEAn`^KS9\3E9]?JQ1K2<S3E9

(x∗, λ∗)9@7FATKS0e<fQ1Y60^A7:AD/4A=Y6Q10S0>/4Y6C=9_L89 L LS/1027 Rn × Rq Z

K2/102Lq3E9@7#Q102?JA=Y6Q1027(90F9K7=Q10^A(L89K8uQ1Yc7(L8YEfIUC=9U0^A=Yc/4jS369U7UlOQ10q<>9UK8A.I@?JCDYECD9nKS0S9;?JQR02L8YEA=Y6Q10q0SIU?9U7D7=/1YECD9LSKLS9K8u8Y6iWX9Q1C[L8CD91ZV,!/r7=Q1K27:b W/4A=CDY6?9qL8K M9U7D7:Y690L8K ,!/1h1C[/40ShRYE9U0 L /17D7=Qg?YEIU9q/1K8uL8IUC=Y6N1IU9U7_7=9U?JQR02L89@7<>/4CkC[/4<S<fQ1C=Anox9@7FA

L∗ = ∇2f(x∗) +

q∑

j=1

λj∇2hj(x∗).

Page 88: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

O % > 4 K \ K K 4 ! & Y

x∗ 9U7:A;WXYE02YEWKSW36Q8?/43 lS/436Q1C[7T36/qW/OADC=Yc?J9L∗ 9U7:AkA=936369+`RK29

dT L∗d ≥ 0, ∀d ∈M. - 9U7:A#3c/MCD9U7:A=CDYc?BA=Y6Q10+LS9L∗ /1K7=Q1K27:b 9U7=<2/1?9 M

`^KSY8L8Q1YEAK A=CD9G7=9WXYEb L8I20SY69G<fQR7=YEA=Y6N191Z 3g0y- 9U7:A#<2/R7!02IU?J9@7=7D/4Y6CD9`^KS9L∗ 369+7=Q1YEAUZ

![ l S!`& !`"+ *+|\0r?JQ10>7:YcL8iCD9_3E9_<SCDQ1j23EiUWq9

(PI)

YE0SY6WXY67=9Cf(x)7=Q1K>7gi(x) ≤ 0, ∀i = 1, . . . , p.

,P.-UKMLB8MOSVU )132 SVU B8C:QWMNU B%@)QWAB%M @ Kj=vvq5m^a^5 CDJ8KL?3@F=:AaKL?Q5b=`CI?[A[t:5 J`7 Em=`?c7@DH5 5mK x^5FEkCDJ8K

?3@F=:AaK)?5FEM*J:KY? ^5FEAaKMOA[CF5DEpEFJ8KL?GM]=:K(E

I(x) = i ∈ 1, . . . , p | gi(x) = 0 .i C:S ? SV=%MLB%MNSTU )1 A

x∗ 5FEI?'EFJ8^P7?3AJ8K JDvq?[A_;b=^f5^J`CF=:^5 M*5 (PI) x∗ 5FEI? EmJ:^[7(?3AJ8K JDvq?[A_;b=:^5^J`CF=:^5 M*5 A_K(A_;yAoEm5F@

f(x)EmJ`7OEgi(x) = 0, ∀i ∈ I(x∗).d\Q102?_369U7k?Q10^A=C[/4Y60^A=9U7TY602/1?JA=Y6N19@7G0!- QR0RAM/1K2?JK20S9_YE0 >KS902?9+7:K2Ck3H- Q1<8ADYEW/13EYEA=I+L8Kr<SCDQ1j23EiUWq9

(PI)Z

PS C F @ 1 K@ H U H A V@C J`7OEq^m\ .vNJ`?! YeDEm5lM*5pMOA)(HF@F5FKY?[Ao=(n:Aa^_AP?QH58?NM*5S@DHBdY7^o=@cAP?QHM*5FEGCDJ8K?3@F=:AaK)?5FE>=*C ?3APt85FE ^5FE<d]@m=`MOA5mKL?[E ∇gi(x

∗) i ∈ I(x∗)EmJ:KL?^aA_KYH=:Aa@F5F;s5mKL?AaKM*HvN5FKM]=KY?PE EA x∗ 5FEI? 7K

;yA_K(A_; 7; ^fJCm=^vNJ`7@g^5uvL@DJ`n:^em;s5(PI) =:^J8@mEuA_^l5B9`AoE?5 p

;s7^P?3A vL^aA3CF=`?Q5:7@mE>M*5y7 (K 7`C`5m@µi ≥ 0

i = 1, . . . , p ?5m^oEuh`7]5

∇f(x∗) +∑

i∈I(x∗)

µi∇gi(x∗) = 0, ! Z#"87D$

µi ≥ 0, i = 1, . . . , p ! Z "D"%$µigi(x

∗) = 0, i = 1, . . . , p. ! Z "+$|\j27=9CDN1QR027M`^KS9X369U7]IU`^K2/4A=Y6Q1027 ! Z "D"%$ b ! Z#"@+$ML8YEA=9U7_L89?Q1WX<S36IWX90^AD/1C=YEA=IX7=YEhR0SY>90^A_`^KS9X369WKS3EA=Y6<S3EYEb?/4A=9UKSCµi/17D7=Qg?YEI]o+KS029]?Q10^A=C[/4Y60^A=9;Y602/1?JA=Y6N19\9U7:AG0gKS3 ZS|\0aC=9U0^NRQ1Y69MogYE0SQRK8u 5 9V<>QRKSCG3c/zLSIWXQ1027:A=C[/OADYEQR0L8K&A=w2IQ1CDiWX9m`^KS9m3 - QR0pY63E36K27:A=CD9C[/z<>/4CkKS0r9Ju89UWq<23E9RZ

@F ?KJO@ )1 J:A[? ^5vL@DJ`n:^em;s5 AaKA_;yAoEm5F@

f(x) = (x1 − 2)2 + (x2 − 1)2EmJ`7OE x2

1 − x2 ≤ 0x1 + x2 − 2 ≤ 0−x1 ≤ 0. \ 5mK Em5F; n:^5

ΩM*5FEyEFJ8^P7?3AJ8K(E @DHD=:^aAoEB=(n:^f5DE<5DE?@F5vL@DHFEF5mKL?QHbE7@<^f= `dY7@D5 % K = EFJ8^P7?3AJ8KWJvq?3Aa;b=:^5 5DE?

^f= vL@FJ D5 C ?3AJ8K0M 7 vqJ8AaKL?(2, 1)T E7O@ CF58? 5FK(EF5m; n^f5 \ 5DE?<M*J:KCb^5yvNJ:A_KL?

x∗ = (1, 1)T Cc@BA[t:J8K(Ey^5FECFJ:KMOAP?3AJ8K(EM`5y7 K 7C*5m@5FK

x∗ V 587^5FE^5FE6M*587:9yvL@F5F;yAfeF@F5DECFJ8KL?[@m=A_KL?Q5DEEmJ:KY?'=`CI?3APt85DE 2(x1 − 2) + 2µ1x1 + µ2 = 0

2(x2 − 1)− µ1 + µ2 = 0. K J`n`?[Af5FKL?!M`J8KC

µ1 = µ2 = 2/3

Page 89: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

-KK

-

6

j

-

g3 = 0

g2 = 0

g1 = 0

x∗

∇g2(x∗)

∇g1(x∗)

−∇f(x∗)Ω

O

Z & QR3EK8ADYEQR027]C=I@/436Y67D/4j23E9@7\9A+7:QR3EK8ADYEQR0QR<8A=Y6W/4369 V369h1C[/1LSYE9U0RA ∇g2(x∗)9JAmQ1C=A=wSQRh1QR02/43/4K<S36/10A[/40ShR90^A;oq36/X?JQR0^A=C[/4Y60RAD9

g2(x) = 090

x∗ L89mWX WX9m<fQ1KSC ∇g3(x∗)Z

X en 'G)'G @C:A+MNAD@ )132 & Q1YEA

f3c/qQ102?JA=Y6Q10rL89

R2 LS/1027

RL8I20SY69m<2/4C

f(x, y) = 100(x2 − x21)

2 + (1− x1)2.

Q10^A=CD9C]`RK29x∗ = (1, 1)

9U7:A\KS0<>QRYE0^A]7:AD/4A=Y6Q10S0>/4Y6C=9zLS9fZ /43c?JKS369C ∇2f(x∗)

9A_LSQ10S0S9UC\3c/02/4A=KSCD9LS9x∗ Z @C:A+MNAD@ )1 & Q1YEA

f3c/X0SQ1CDWX9\9UK2?J36YcL8YE9U0S0S9+7=KSC

Rn L8I 202YE9RlS<>QRKSC x ∈ Rn lS<2/1Cf(x) =‖ x ‖2=

(x1)2 + · · · (xn)2.

"RZ Q10^A=CD9Ck`^KS9f/RL8WX9JAkKS0S9+LSICDYENRI9mL8Y6C=9@?BA=Y6Q1020S9363E9_9U0aADQ1K8A;<fQ1Y60^AML89

Rn \ 0 9AM`^KS9

f ′(x; d) =1

‖ x ‖2xT d.

SZ |\0r7=9]<236/R?J9mLS/40>7RZSx;36Q1C[7

f(x) = |x| 0y- 9U7:Ak<2/17kLSICDYENO/4j23E9_90 0Z>|\0rLSI 20SYEA

fε(x) =

|x| − ε2 si |x| > ε

12εx2 sinon

A=K2L8Y69CM36/XL8IUC=Y6NO/4jSY63EYEA=I+LS9 fεZ KS936369]9@7FA;3 - 9Ju8<SCD9U7D7:Y6Q10&L89

fε<>QRKSC

f =‖ · ‖27:KSC

Rn Z @C:A+MNAD@ )1 |\0r?Q1027=YcL8iCD9\3c/Q102?JA=Y6Q10`^K2/1LSCD/4A=Yc`RK29]L8I20SY69+LS/40>7

R2 <2/4Cf(x, y) = 5x2 + 5y2 + 8xy − 10x− 8y.

Page 90: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

7 % > 4 K \ K K 4 ! /D$GQR0^A=CD9C;`^KS9

f9@7FA;KS029_Q102?JA=Y6Q10r?JQ10gNR9Ju89\9AML8IJAD9CDWXYE0S9UC;7:QR0pWXY60SY6WzKSW h136Q1j2/13yLS/4027

R2 ZjV$ 9U<SCDIU7=90^A=9UCT369U7M?JQ1K2C=jf9U7nL89m0SY6N19@/4KrL89fZ?8$ex<2/4C=A=Y6CeL8K <fQ1Y60^AaYE02Y~ADY6/13

(−4, 4)l./4<S<23EYc`^KS9CaYEA=IUCD/4A=Y6Q1027LS9U7eWXIA=wSQ8L89@7eL8K h1C[/1LSYE9U0RAe9AaL8Kh1C[/1LSYE9U0RA;?Q10 FKShRKSI1Z QRWX<2/4CD9Cn369U7nCDIU7=KS3~A[/OA[7nQ1j8AD90gK27UZ

@DC:ADMNA+@ 1 & Q1YEA;36/Q10>?BA=Y6Q10fL89

R2 LS/4027

RL8QR0S0SI9_<2/1C

f(x) = x21 + 2x2

2 + 4x1 + 4x2.

QR0RADC=9UCk<2/4CkCDIU?KSCDC=9U02?J9_`^KS9m36/XWXIA=wSQ8L89+L89m3c/X<S36K27nQRC:AD9m<>9U0RAD91l2/NR9U?dk = −∇f(xk)

l>/4<S<S36Yc`RK2I9+of90r<2/1C:A[/40^A;L89

(0, 0)hRI0SiUC=9]KS0S9+7=KSYEA=9mN19@?BADQ1CDYE9U3E369 xk

LS9\AD9CDWX9mh1I02IC[/43

xk =

(

23k − 2

(

− 13

)k − 1

)

(0L8I@L8KSY6C=9_369mWXYE0SY6WzK2W L89fZ

@DC:ADMNA+@ 1f/ & Q1YEAf : Rp −→ R

L89KSupQRY67\L8Y~ICD90^ADY6/1jS3E9z9JAAK20S9+W/OADC=Yc?J9

p× nL89CD/10Sh

pZ & Q1YEA

q : Rn −→ R

LSI 20SY69_<2/4Cq(x) = f(Ax)./D$ /136?KS3E9UCk3E9mhRCD/RL8Y690^AT9A;3E9 M9U7D7:Y690L89q90&Q102?JA=Y6Q10rL8Krh1C[/1L8Y690^An9JAML8K M9U7D7:Y690L89

fZjV$nxM<S<S36Y6?U/OA=Y6Q10 `(CDQ1j23EiUWq9+LS9U7nWXQ1Y602L8CD9U7k?U/4CDC=I@7Z

q(x) =‖ Ax− b ‖2,Q+,

b9@7FAkKS0pN19@?BA=9UKSCkL89

Rp L8Q1020SI1Z /43c?JK23E9UC x∗ `^KSYVWXYE0SY6WXY67=9 q

9JAkWX9JA=A=CD9x∗ 7=Q1K27T3c/+QRC=WX9 x∗ = A∗b

Z KS9U3E369U7k7=Q10^Ak3E9@7k<SCDQ1<SCDY6IJA=I@7nL89

A∗ Z @DC:ADMNA+@ 1 Q10^A=CD9CM`^KS9qLS/40>7;3c/WXIJA=w2QgLS9L8KhRCD/RL8Y690^AM?JQR0 FKSh1K2Iz/4<S<23EYc`^KSI9zoeKS0S9+QR02?BADYEQR0`^K2/ObL8C[/OADY6`^KS9m?JQR0gN19Ju89Rlg/NR9U?\WXYE0SY6WXY67D/OADYEQR0r9JuS/1?JA=91l8369U7n02Q4AD/4A=Y6Q1027k7=KSY6NO/40^A=9@7k7:QR0RAkNRICDY2IU9U7UZ Q Vk = lind0, d1, . . . , dk = ling0, g1, . . . , gk

dTk+1Adi = 0

l ∀i = 0, 1, . . . , kZ

A tk =

‖ gk ‖2dT

k Adk

@DC:ADMNA+@ 1 d\Q1020S9Ck369U7k?Q1KSCDj>9@7kL89m0SY6N19@/4KrL89_36/qQR02?BADYEQR0f(x) = x2

1 + 4x22 − 4x1 − 8x2.

(0L8I@L8KSY6C=9_369mWXYE0SY6WzK2W L89fZ/D$QR0RADC=9UC\`^KS9q36/pWXIJADwSQ8L89qLS93c/a<23EK27\QRC:AD9<f90^A=9X/4<2<S3EYc`^KSI9Xo

fl9U0<2/1C:A[/40^A]L89

(0, 0)0S9q<>9UK8A?JQR0gN19CDh19UCG9U0rKS0r0SQ1WjSC=9 20SY*LV- YEA=IUCD/4A=Y6Q1027UZjV$a!CDQ1KSNR9CeK20 <>QRYE0^ApY60SYEA=Yc/43]<>QRKSCp`^KS936/ WXIA=wSQ8L89L8K h1C[/1LSYE9U0RAa?Q10 FK2h1KSILS9 #3E9AD?[wS9UC 99UN19@7/4<2<S3EYc`^KSI9+o

f?JQ10gNR9CDh19M9U0rKS0S9+7=9KS369_Y~ADIC[/OA=Y6Q10!Z

@DC:ADMNA+@ 1 & Q1YEA;KS0S9]Q102?JA=Y6Q10`^K2/1LSCD/4A=Yc`RK29f : Rn −→ R

L8I20SY69]<>/4C

f(x) =1

2xT Qx− bT x,

Q+,Q9U7:A;KS0S9_W/OADC=Yc?J9m?/1C=CDI9_7=5^WXIJADC=Yc`^KS9mLV- Q1C[L8C=9

nZ

Page 91: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

-KK ""RZ d\Y67D?JK8AD9CmL89q3H- 9Ju8Yc7FAD902?99A+L893c/e0>/OA=K2C=9XL89U7]<fQ1Y60^AD7_7FA[/OADYEQR0S02/4Y6CD9U7MLS9

f9U0QR02?BADYEQR0L8KCD/10ShLS9m36/qW/OADC=Yc?J9

QZSZ & Q1YEA

xK20<>QRYE0^A\7:AD/4A=Y6Q10S0>/4Y6C=9Rl

λjKS0S9zNO/13E9UKSCM<SCDQ1<2C=9+L89

Q9JA

uj3E9N19@?BA=9UKSC;<SCDQ1<2C=9z/R7=7=Q8?JY6I+o

λjZ(QR0^A=CD9Ck`^KS9

f(x + tuj) = f(x) +1

2t2λj , ∀t ∈ R.

2Z d\Y67D?JKSA=9C(LS9T3c/\h1IQRWXIJA=CDY69TL89@7?JQ1K2C=jf9U7L89n0SY6N19@/4Kz9A(L89n3H- I<SY6h1C[/4<2wS9TLS9fLS/4027#369U7P?U/177:K2YENO/40^A[7

s2 ,!/qW/OADC=Yc?J9Q9U7:A;L8I 202YE9m<fQR7=Y~ADYENR91Z

,!/qW/OADC=Yc?J9Q9U7:A;7:IUWqY LSI 20SY69_<>Q^7:YEA=Y6N19m9AnY63y9ugYc7:A=9mL89U7;<>QRYE0^AD7k7:AD/4A=Y6Q10S0>/4Y6C=9@7Z

,!/qW/OADC=Yc?J9Q9U7:A;7:IUWqY LSI 20SY69_<>Q^7:YEA=Y6N19m9AnY63y0y- 9Ju8Yc7FAD9m<2/17kLS9]<fQ1Y60^AD7;7:AD/4A=Y6Q10S0>/4Y6C=9@7Z

,!/qW/OADC=Yc?J9Q9U7:AkYE02LSI 20SY69m9JA;0SQR0&7=Y60Sh1KS36Y6iCD91Z

QRWXWq9m/1<S<S36Y6?U/OA=Y6Q10!l8Q10&<>QRKSC=C[/qK8A=Y636Y67=9Ck369U7nW/OADC=Yc?J9@7Q9A;3E9@7kN19@?BA=9UKSC[7

b7=KSYENO/10RA[7lSL2/4027

R2

Q =

(

5 44 5

)

, b =

(

918

)

cas (C1)

Q =

(

−2 22 −2

)

, b =

(

1−1

)

cas (C2)

Q =

(

−2 22 −2

)

, b =

(

10

)

cas (C3)

Q =

(

−2 22 6

)

, b =

(

6−6

)

cas (C4)

Page 92: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

% > 4 K \ K K 4 !

Page 93: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

o~ m

|\0 IJADK2L8Y69LS/1027r?J9?[w2/4<2Y~ADC=9L89U7&WXIJADwSQ8L89U7rYEA=IUCD/4A=Y6N19@7a<fQ1KSCrCDIU7=Q1K2LSC=9K20 7=587FADiWX9Ly- I@`RK>/OA=Y6Q10>736Y60SIU/1YECD9U7UZ 3S7- /4hRY~APL89nWXIJA=w2QgLS9U7PL89nAF5g<f9T<fQ1Y60^A Su89n`^KSY89U0Sh19U02L8CD90^AKS0S9k7:K2Y~AD9nLS9n7=Q136K8ADYEQR027P`^KSYS?JQR0^NR9CDh19NR9C[736/m7=Q136K8ADYEQR0XLSKe7:587:A=iWX9RZ^|\0N19UC=C[/\`^KS9\?[w2/1`^KS9nYEA=IC[/OADYEQR0X<>9UK8AG7U- 9f9@?BA=K29C.9U0O(n2) ! 9U0 O(n)

<fQ1KSCLS9U7(W/4A=CDY6?9U7Pj2/102L89@7 $9AG`^KS9M?9U7PWXIJADwSQ8L89U7.<>9UKSN19U0^A JADC=9M?Q1WX<fIJA=YEA=Y6N19@7./NR9U?T3E9@7(WXIJADwSQ8L89U7GL89;<SY6N1Q1AD/1h197=Yn9U7:ATADC=i@7kh1C[/40>LVZ

a T&"$p $p q+'T"&)& QRY~A

Ax = bK20a7:587:A=iUWq9\3EY60SI@/4Y6C=9MLS9;W/OA=CDYc?J9A ! n×n

$(0SQ10e7=YE02h1KS36YEiUC=9\L8Q10^AG3 - KS0SYc`RK29\7=Q136K8A=Y6Q109@7FAx∗ ZS|\00SQ4AD9 x

(k)i3c/

ib iWX9_?JQRWq<fQR7D/40^AD9]L8K&N19@?BAD9KSC

x?/136?KS3EI_oq3H- YEA=IC[/OADYEQR0

kZ2|\0r7:KS<2<>Q^7:9UCD/z3E9@7nI36IWX90^AD7

aiiL8Yc/4h1QR02/4KSuLS9

AA=QRK27]0SQ100gKS367UZy,!/aWXIJA=w2QgLS9X?Q1027=Yc7FAD9qo&?/136?KS3E9UC]?[w>/1`^KS9zNO/1C=Yc/4jS369X7:IU<2/4CDIWX90^A@l>369U7_/4KSA=CD9U7\IJAD/10^A

2ugIU9U7koq369KSCkNO/4369K2C;L89m3H- YEA=IC[/OADYEQR0p<2C=I@?JIULS90^A=9RZ" J TS C:MNB F @ K@QWADS Mk = 0 x(0) ∈ Rn LSQ10S0SIk ≥ 1

Q1KSCi = 1, . . . , n

lS?U/43c?JKS369C

x(k+1)i =

1

aii

[

bi −n∑

j=1,j 6=i

aijx(k)j

]

,*/]?Q10gN19UC=hR902?9.L89n3c/\WXIA=wSQ8L89n0SI@?J9U7D7=Y~AD9nLS9U7wg5g<>Q1A=wSi@7:9@7#7=KS<S<23EIUWq9U0^AD/4Y6CD9U77:K2CA?Q1WXWX9T369nWXQ10^A=CD93 - 9Ju89WX<S369+7:K2YENO/40^A

[

1 1010 2

] [

x1

x2

]

=

[

1112

]

LSQ10^Ak36/7=Q136K8A=Y6Q10&9U7:Ax∗ = (1 1)T Z wSQ1Yc7=Y67D7:QR027 x(0) = (0 0)T 23E9@7nY~ADICDIU7kL89^/1?Q1jSYV7=Q10^A

x(1) =

[

116

]

, x(2) =

[

−49−49

]

, x(3) =

[

501251

]

, x(4) =

[

−2499−2499

]

9A;36/qWXIJADwSQ8L89+L8Y6N19UC=hR91Z9CDWzKSA=Q10>7k/436Q1C[7G369U7ML89K8u&?JQ136Q1020S9U7kL89

A9A;Q10&Q1j8ADYE9U0RA

x(1) =

[

1.21.1

]

, x(2) =

[

.98

.98

]

, x(3) =

[

1.0041.002

]

, x(4) =

[

.9992

.9996

]

Page 94: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

4 K 9JA;3c/qWXIJA=w2QgLS9+?JQ10gNR9CDh191Ze[ Tp"$&$p+%p*k)J$pk

,y9@7kWXY67=9U7\o FQ1K2CM7:QR0^AM9J9U?JA=KSIU9U7M7=IU`^KS9U0^A=Y69363E9UWq9U0^A;90<SCD902/10^AM<>QRKSCxjlj < i

l23c/L89CD0SY6iCD9mN/13E9UKSC?/136?KS36I9m7:QRY~A

x(k+1)j

Z" J VSVC:MLB F @K@ QWHK=8= @M K@Jk = 0 x(0) ∈ Rn L8QR0S0SIk ≥ 1

Q1KSCi = 1, . . . , n

l2?/136?KS369C

x(k+1)i =

1

aii

bi −i−1∑

j=1

aijx(k+1)j +

n∑

j=i+1

aijx(k)j

@F QWC HK@ 132 =*h`7]5AP?QHF@m=?3AJ8K ;yAaEF5 DJ*7@uM`5FE nCFJ8; vqJ:Em=:KL?Q5DE E*\ 5 (58CI?c7]565mK O(n2) ! 4 vqJ*7@^5FE>M`587:9 ;bH:?! YJM*5DE \ A_; vL^Hm;s5FKY?Q=`?[AfJ:KjM*5<^F\ AP?QHF@m=?3AJ8KWM*5 =`7.EcE 5mA[M*5m^'KL5<KYH CD5FEBEAP?Q5sh`7]5y^f5<EI?QJC=8d5

MN\ 7K Em5:7^ ?P=(n:^5D=`7 M]=K(E^f58h`7]5m^)^5FExi

EmJ:KL?N;yAoE DJ*7@pEFH h`7]5FKY?[Af5F^_^5m;s5FKY?'=^fJ:@BEh`7]5 M*5879 ?P=(n:^5D=`79sEmJ:KY?KYH8CF5FEBEm=:Aa@F5DEvNJ`7@ =*CDJ`n:A \ AaKL?QHm@Dr8?'M*5CF5:??5M*5F@cKAem@D5!;sH8?% YJM*5@DHFEcA3M`5qvL^[7(?*?SM]=:K E'^5 3=:AP?'h`7]5G^5FES;yAaEF5FE DJ*7O@uM*5FECDJ8; vNJEB=KL?Q5FEM*5

xvq587(t85mKL?'E*\ 5 (58CI?c7]5m@5FK>v=:@F=:^a^feF^f5E7O@uM*5FE'vL@DJCF5DEcEF587@mEMOA)(HF@F5FKY?PE

@F QWC HK@ 1 K vN5:7?qH8Cc@BA_@D5M*5DEpt:5m@mEAJ8K(ElM*5FE=:^_d`J:@cAP?! ;s5DEM*5 =*CDJ`n:ALJ*7 =7OEcE 5mA[M*5m^]v)=:@n:^fJCmE C =`h*7*5

xi5DE?G7K n:^fJCM*5uCDJ8; vqJ:Em=:KL?Q5DE?5m^Nh`7]5l^f= EmJ`7OE 3;b=`?[@cA[CF5M.Ao=:d`J8KY=:^5

aii5FEI?NKYJ8KsEcAaK`d 7^aAem@D5 KY5

;yAaEF5 DJ*7@uM`5 ? Ovq5 =*CFJn:AL@F5:t8Af5FKY?'=^fJ:@BE @FHDEmJ`7`MO@D567OK E (E?em;s5^_AaKYH=:Aa@F5 M*5ovq58?[A[?5 ?P=A_^a^f5

[aii]x(k+1)i = bi −

n∑

j=1,j 6=i

[aij ]x(k)j .

e/. "+ +n 1+kp'T$pk Tp"$&k )[*n _*) +k%$l W+ N ')&

|\0L8Q10S029ADQ1K8A+LV- /4jfQ1C[LKS0CDIU7=KS3EAD/4AmLS9X?Q10gN19UC=hR902?9z7=KSCm3E9@7_<SKSYc7=7D/40>?J9U7_L89XW/OADC=Yc?J9@7_`RK2Y(<SCDIU?JYc7=9369mC=R3E9mL89@7nN/13E9UKSC[7n<SC=QR<SCD9U7UZ>|\0pC[/4<S<f936369_`^KS9ρ(A)

9U7:A;3E9mC[/51QR0p7=<>9@?BA=C[/43yLS9]3c/XW/OADC=Yc?J9AlSY Z 91ZS369m<S36K27h1C[/40>LaWXQ8L8KS369+L89U7nNO/13E9UKSCD7n<2C=QR<SC=9@7nL89

AlSNRQ1Y6C;?[w2/4<yZ Z

PS C F @ )1[2 J8AP?l7KY5;b=`?[@cA[CF5 Cm=@c@DH5B ^5FECDJ8KM.A[?[AfJ:K(ElE7OA[tD=:KL?5FElEFJ8KL?SH h`7APtF=^f5FKL?Q5FE

A limk→+∞ Bk = 0

A_A limk→+∞ Bkv = 0 ∀v ∈ Rn

A_AaA ρ(B) < 1

Q1KSCk3c/XL8IWXQR027FADCD/4A=Y6Q10yl8NRQ1Y6C#Z 3qZ Yc/4CD3E9A-5 SlS<yZ " 9 Z PS C F @ )1 J8AP?l7KY5;b=`?[@cA[CF5

B n× n KLJ8K EAaK`dY7^_Aem@D5 7OKyvL@DJCF5DEcEI7OEAP?QHm@F=`?[A SM]=:K E Rn

u(k+1) = Bu(k) + b

M*HKA 6v=:@ ?3Aa@uMN\ 7OKsv)=:@ ?3Aa@6MN\ 7KsvqJ8AaKY?u0 58?!E7Iv(vNJEmJ:K(Eh`7\ Aa^YvqJ:EBEFe M*5 7KbvqJ8AaKL? `9 5 C`\ 5DE? M.A_@D5 h`7AEm=`?3AoE 3=:AP?

u∗ = Bu∗ + bW 5GvL@DJCF5FEBEI7OECFJ8KLt:5m@cd`5lEcA 58?'EF587^5m;s5FKY? EcA ρ(B) < 1

Page 95: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W W >K 4 4 K K ^wyxz|~L*~L|0

e(k) = u(k) − u∗ = Bke0A=9U02LaNR9C[7

07=Y

ρ(B) < 1Ly- /1<[email protected]]A=wSIUQ1CDiWX9 Z "RZ

6@F QC" H @ 1 J8;y;s5p^f5p@m=(J8KbEQvN58CI?[@m=^)5DE?qEFJ*7t:5mKL?SM.A kCcAa^5 yH8tD=:^P7]5F@ ^f=>CDJ8KMOAP?3AJ8KsKYH CD5FEBEB=A_@D55:?EI7>Em=:KL?Q5 M(7 ?! LHJ8@Dem;s5 vN5:7? Qr8?[@F5@D5m; vL^f=*CDH5Gv)=:@l^f=sCFJ:KMOAP?3AJ8K E7>Em=:KL?Q5

‖ B ‖< 1J ‖ · ‖ 5DE?l7KL5pKLJ8@B;b5EI7(n`J8@IM*J8KKLH5 h`7]5m^[CFJ:Kh`7]5 l%$ SS")& '"L" "+ p "+ "&(" + N` S

,!9U7WXIJADwSQ8L89U7XYEA=IUCD/4A=Y6N19@7IA=K2LSYEIU9U7eLS/1027?J9?[w2/1<SYEA=CD9r?Q1CDC=9@7:<fQ10>L890^AoK20S9L8IU?Q1WX<>Q^7:YEA=Y6Q10L893c/W/4A=CDY6?9AL8K 7:587:A=iUWq936Y60SIU/1YECD97:QRK273c/Q1CDWX9

A = M − N/N19@?

M0SQR0 7:Y60Sh1K23EY6iCD91Z(,!97:587:A=iUWq936Y60SIU/1YECD9_<>9UK8AM/436Q1C[7n7- IU?C=Y6C=9_?JQRWXWq9_KS0r<fQ1Y60^ASu89

x = M−1Nx + M−1b.d- /4<SCDiU7q369aADwSIQRC=iUWq9r8l3c/WXIJADwSQ8L89rY~ADIC[/OADYENR9p7=9C[/?JQR0gN19CDh19U0^A=9a7=Yn3E9rC[/51Q10 7=<f9U?BADCD/13kL89p3c/WX/4A=CDY6?9M−1N

9U7:AM7FADC=Yc?BAD9WX90^AkY608ICDYE9UKSC;o"1ZQ1K2C9ug<23EYc?JYEA=9UCz3c/L8I@?JQ1WX<fQR7=Y~ADYEQR0 ?JQRC=CD9U7=<>QR02LS/10RAz/1K8uWqIA=wSQ8L89@7XL89 ^/1?Q1jSY(9AG3_/1K27D7Fb & 9YcL89U3Hl*Q1002Q4A=9

A = A − E − FQ+,

D = diaga11, . . . , a229JA −E

9JA −F7=Q10^AM369U7\<2/1C:ADYE9@7ML89

AA=CDYc/40ShRKS36/1YECD9U7CD9U7=<f9U?BADYENR9WX90^A;/4KrL89@7=7=Q1K27n9A;/4KL89U7D7=K27kL89_36/LSY6/1h1Q10>/4369 ! ?JFZ 2hRKSCD9 Z " $BZ>|\0&Q1j8ADYE9U0^AM/436Q1C[7 % : '8 2 8 2 q ' ; > M = D

lN = E + F ! QR0r0SQ4AD9C[/ J = D−1(E + F ) % : '8 2 8 2 s 4 4 " 2 ;8 2.o > M = D −E

lN = F

Z

−E

−E

DA=

Z#" d\IU?JQRWX<>Q^7:YEA=Y6Q10L89_3c/XWX/4A=CDY6?9\<fQ1K2C;3E9@7nWXIJA=w2QgLS9U7;L89^/1?Q1jSYV9AML89 3_/4K27D7Fb & 9UY6L89U3

|\0<f9KSA]/1?U?JIU3EIUC=9UC;?J9U7]L89UK8u&WXIA=wSQ8L89@7M9U0YE0^A=CDQ8L8KSYc7=/10^AMKS0 q 2 q5;2=:892 .)2@o : ; ωZ & Y

x(k+1)9@7FA;3 - Y~ADICDI_?U/43c?JKS36I_<2/4CkK20S9+L89U7kLS9K8u&WqIA=wSQ8L89@7l8QR0?/43c?JK23E9UCD/x(k+1) = ωx(k+1) + (1− ω)x(k)..0X/4YEAUl1QR0KSA=Y63EYc7:9UCD/m?9JA=A=9;YcL8I9;7=KSC=A=QRK8A./N19@? 3_/1K27D7Fb & 9YcL89U3HZR|\0XQ1jSA=Y690^A(36/_WXIA=wSQ8L89MLSY~AD9ML89kCD93c/OuS/OADYEQR0yZ,*/XL8IU?Q1WX<>Q^7:YEA=Y6Q10r?JQRC=CD9U7=<fQ102LS/10^A=9M9@7FA

M =1

ωD −E, N =

1− ω

ωD + F.

Page 96: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

+ 4 K ,(- Q1<fIC[/OAD9KSC#L89.<fQ1Y60^A Su89.9@7FA#0SQ4ADI

Lω = (D−ωE)−1[(1−ω)D+ωF ]ZL19U7:A#LSQ102?P3 - Q1<fIC[/OA=9UKSC*LS9.<fQ1Y60RA

Su89+L89m3c/XWXIJA=w2QgLS9zL8943_/4K>7=7:b & 9YcL893 Z>|\0r<2/4CD3E9UCD/qL89 4 vs 4 " .)2.o :<; 7=Y ω < 19JAMLS9 4Vs . " .)2@o : ; 7=Y

ω > 1Z

PS C F @ )1 7IvvqJ:EFJ8K(EAE ;bH:?3@BA3h`7]5u5:?N^f=sM*H CDJ8; vqJ:EcA[?[AfJ:K

A = M −N?Q5F^_^5 h*7*5

MT + NEmJ:A[?

M*HKA5vNJEAP?3APt85 ^J8@mEρ(M−1N) < 1

EA 58?'EF587O^f5F;b5FKL?'EA A 5FEI?M*H K(Af5vqJ:EcA[?[A[t:5 wyxz|~L]~| ; & KS<S<fQR7=Q10>7k`RK29 A

9@7FAMLSI 20SY69m<fQR7=Y~ADYENR91Z>|\0&Q1j27=9CDN19_LV- /4jfQ1C[Lp`^KS9MT + N = (AT + NT ) + N = (A + N) + NT = M + NT .

(KSYc7D`RK293E9C[/51QR0v7:<f9U?JA=C[/43k9@7FAejfQ1CD0SI7:K2<>IUC=Y69KSCD9WX9U0RA<2/1CXA=QRK8A=90SQRC=WX9rW/OADC=Yc?JY69363E9RlPQ10sNO/INO/436KS9C‖M−1N ‖ /NR9U?\36/q0SQRC=WX9m7=KSj>QRCDLSQ10S0SIU9]oX3c/q0SQ1CDWX9_N19U?JA=QRC=Y69363E9 ‖ v ‖A= (vT Av)1/2

‖M−1N ‖=‖ I −M−1A ‖= sup‖v‖A=1

‖ v −M−1Av ‖A .

QR7=Q1027w = M−1Av

Z>x;36Q1C[7‖ v − w ‖2A = (v − w)T A(v − w)

= 1− wT Av − vT Aw + wT Aw

= 1− wT Mw − wT MT w + wT Aw

= 1wT (MT + N)w.|\0L8IULSKSY~AM/13EQRCD7n`^KS9 ‖ v − w ‖2A< 1hRCR?J9_o3 - w^5g<fQ4ADwSiU7=9m7:KSC

MT + N ! <fQ1K2C w 6= 0$JZ

; ; & KS<S<fQR7=Q1027p`RK29ρ(M−1N) < 1

Z & YA0y- 9U7:Ap<2/R7pL8I20SY69<fQR7=Y~ADYENR91lGYE3_9Ju8Yc7FAD9KS0

x0A=9U3]`^KS9

α0 = xT0 Ax0 ≤ 0

Z ADK2L8Y6Q1027;36/7=KSYEA=9 αkL8I20SY69m<2/4C

α09JAk<2/1C

αk = xTk Axk

9JAxk = Bxk−1

Z 9JA=A=97=KSY~AD9_A=9U02L&N19C[77X?/1Cρ(B) < 1

Z Q1WXWX9xk−1 − xk = M−1Axk−1 = N−1Axk

lSQ10r/ αk−1 − αk = (xk−1 − xk)T MT A−1M(xk−1 − xk)− (xk−1 − xk)T NT A−1N(xk−1 − xk)

= (xk−1 − xk)T (MT + N)(xk−1 − xk)?/1CA−1M − I = A−1N

9JANT A−1 + I = MT A−1 Z>|\C xk−1 − xk 6= 0

7=Y60SQ10&Q10r/4KSC[/4YEABxk−1 = xk?J9q`^KSY*Y6WX<S36Y6`^KS9UCD/1Y~A_`RK29

B/aKS0S9N/13E9UKSC]<SC=QR<SCD9mIhR/13E9o)"1l?J9q`^KSY*9@7FA]YEWX<fQR7D7:Y6jS369z<SK2Y67D`^KS9

ρ(B) < 1Zd\Q102?z3H- wg5^<fQ4ADwSiU7=9z7=KSC

MT + NY6WX<S36Y6`^KS9q`RK29+36/a7:KSYEA=9 αk

9U7:A\L8I@?JCDQ1Yc7=7D/40^AD9]9A]?Q1WXWX9α0 ≤ 0

l>YE3#5/?JQR0^A=C[/1L8Yc?BADYEQR0p/N19@?M369]/4YEAM`^KS9+?J9A:A=9+7=KSYEA=9m?JQR0gN19CDh19]N19UCD7 72ZSdMQR02?A9U7:A;L8I 202YE9m<fQR7=Y~ADYENR91Z

Z

S C:STJNJOQWMLC:@ 132AH8?P=KL?LE ;sH8?3@BA[h*7*5lM`H KA5NvNJEAP?3APt85 ^o=;sH8?% YJM*5pM*5S@F5F^o=:9(=`?3AJ8KsCFJ:KYt:5m@cd`5'EA 58?EF587^5 ;s5mKL?'EcA 0 < ω < 2

wyxz|~L]~| V|\0&INO/436KS9m7:Y6WX<S369WX90^AMT + N

MT + N =1

ωD −ET +

1− ω

ωD + F =

2− ω

ωD.

,y9@7yW/OADC=Yc?J9@7V0SQR0z7:5gWXIJADC=Yc`^KS9U7!L8I 202YE9@7!<fQR7=Y~ADYENR9U7yLSQ10S0S9U0^A*36YE9UKmIhR/13E9UWX90^A*o;L89@7yCDIU7=KS3EAD/OA[7yL89T?JQ10gNR9C=bh19U02?J9m7=Yy369KSC;L8Yc/4hRQ102/13E9]9U7:AM7:A=CDY6?JA=9WX9U0RA;L8QRWXYE02/10^A=91Z PS C F @ )1 A

A5DE? 7OKY5G;b=`?[@cA[CF5 <M.Ao=:d`J8KY=:^5SEI?3@BA[CI?Q5F;s5mKL?qM*J8;yAaK =:KL?5 ^f=u;sH8?% YJM*5M*5 =*CDJ`n:AY5DE?CFJ:KLt85m@cd`5FKL?Q5

wyxz|~L]~| ;M0S9qWX/4A=CDY6?9qo&L8Yc/4hRQ102/13E9z7:A=CDYc?BA=9UWX90^A_LSQ1WXYE0>/40^A=9X7=/4A=Yc7F/1Y~A]jSY690aii > 0

l ∀i Z,!/qW/OADC=Yc?J9J = D−1(E + F )

7D/OA=Yc7:/4YEAJii = 0, Jij = −aij

aii

<fQ1KSCi 6= j.

& YA9@7FAMoXL8Yc/4hRQ102/13E9_7:A=CDY6?JA=9WX9U0RA;L8QRWXYE02/10^A=91l

aii >∑

j 6=i |aij |l>L8Q102? ∑

j |Jij | < 1l2LV- Q+,

ρ(J) < 1Z

Page 97: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

W W >K 4 4 K K

l%$%$ p ! + L G" "! p" &" + N S - 9U7:AkKS0S9m`^KS9U7:A=Y6Q10rL8Y <?YE369moz3c/1`^KS9U3E369]Q10&<>9UK8AkC=IU<>QR02L8CD9]LS/1027n?J9UC:A[/4Y6027n?/17T<2/1C:ADY6?KS36YE9UCD7n?JQRWXWq9]369?U/17kL89@7nW/OA=CDYc?J9U7TADC=YcL8Yc/4hRQ102/13E9@7.<2/1CkjS36Qg?U7Z

PSVCF @ 1o/ J8AP?A7KY5p;>=?3@BA3CD56?[@cA[MOAf=8dJ8K =^f5 v=:@6n:^J`CmEM*H K(Af5!vNJEAP?3APt85 ^J8@mE

ρ(L1) = ρ(J)2 <1 = tD=:^587@Jvq?3Aa;b=:^5 M(7kCFJ*5 kCcA5mKL?M*5@D5m^f=89(=`?[AfJ:K

ωopt =2

1 +√

1− ρ(J)2.

wyxz|~L*~L| PQRYEC Yc/4CD3E9A-5 SlS<yZ "87 9 Z @F ?JN@ )1[2 ^a^P7OE?[@FJ:K(ECF5DEM*5m@BKA5m@mEG@FHDE7^P?P=?[E'v)=@67K 5B9 5m; vL^5

A =

[

2 −1−1 2

]

.

#D H:?! YJM*5gM*5 =*CDJ`n:A

M =

[

2 00 2

]

, N =

[

0 11 0

]

, J =

[

0 1/21/2 0

]

.

5FEutD=:^587O@BE'vL@DJDv@F5DEM`5JEmJ:KL?

1/258? −1/2 ^f= ;sH8?! LJ`M`5gCFJ8KLt:5m@cd`5

2u(k+1)1 = u

(k)2 + b1

2u(k+1)2 = u

(k)1 + b2.

\ 5F@c@D587O@p5DE?M.A[t8AoEmHI5!v=:@ sC =*h`7]5AP?QHm@F=`?[AfJ:K H:?! YJM*5gM*5 =7OEBE 5mA[M*5F^

M =

[

2 0−1 2

]

, N =

[

0 10 0

]

, L1 =

[

0 1/20 1/4

]

.

5FEutD=:^587O@BE'vL@DJDv@F5DEM`5L1

EmJ:KL?05:?

1/4 ^f=<;bH:?! YJM*5gCDJ8KLt85F@d5 2u

(k+1)1 = u

(k)2 + b1

2u(k+1)2 = u

(k+1)1 + b2

=1

2(u

(k)2 + b1) + b2.

\ 5F@c@D587O@p5DE?M.A[t8AoEmHI5!v=:@4sC =*h`7]5AP?QHm@F=`?[AfJ:K

W H:?! YJM*5gM*5 5m^f=89(=`?[AfJ:K

M =

[

2ω 0−1 2

ω

]

, N =

[

−2 + 2ω 10 −2 + 2

ω

]

, Lω =

[

1− ω ω2

ω(1−ω)2 1− ω + ω2

4

]

.

=^3C 7^J8K(E^5GvL@FJM(7OA[?M*5DE tF=^f5:7@mE'vL@FJvL@D5FEpHBd`=:^=`7 M*H:?Q5m@B;yA_KY=:KL? det(Lω) = (1− ω)2.

=ktD=:^587@Jvq?3Aa;b=:^5ωopt

5DE?SEI7IvqHm@BAf5:7@D5 # 58?G5F^_^5 CFJ:@c@D5FEQvqJ8KM>=`7>@F=J8KkEPvq5 C ?3@F=:^);yAaKAa;>=^ M*J8KC^f5DE tF=^f5:7@mESvL@DJDvL@D5FEuM`J8APt85mKL?!r:?3@D5 HBd`=:^5FE

ω − 1 @^587@EmJ:;<;s5<tD=7?

2− 2ω + ω2/4 = 2ω − 2 4k\ J

ωopt ≈ 1.07 5g?P=`79 M*5 CFJ:KLt85m@cd`5FKCF55FEI?S=:^J8@mEpHBd*=^

0.07 ≈ (1/4)2

Page 98: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

O 4 K

Page 99: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

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(C=9@7=7Ul n/43EA=Y6WXQ1CD91l " . Z5 9%$'()|~CBD43|N7(*E, 1/102L%.6GFN~Lz w @ /0 .)2 8". 2Ko ; ".10 : ;6@s=2 s 0 % .1;6@s=2 ZS,!/NRQ1Yc7:Y69C@l" O Z5 9%$H( |79 w "I ; 2 . 8% co ; 2 . . /. 0 0 ; Z(xkLSLSY67=Q10KJ9U7=3E9U51l 9@/1L8Y60Sh2lx+l" O Z5 9g9KSC[/40^A 3qZ @ "0 s : 2 . kows : ; I . 92 C 41: 2 0 4 Z & ADK2L8Y69U7!YE0 /4A=wS9UW/OA=Yc?7!/402L_YEAD7*x;<S<S36Yc?/OADYEQR027UZ;Q1C=A=w;QR3E3c/40>LVl " Z5 9gYE02Q1K8u Z . /. 0 0 : ; : % 0 :<;6.s=2 /032ML Z(dMKS02Q102LVl(/1C=Yc7Ul " O SZ5#"879MN ~L|4 I ; 2 .Oo 92 ' . 8 ; :<4O o ;_q : ;=4 Z TC=QgQ87 Q13691l " O. Z

Page 100: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

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Page 101: ISIMA Analyse Numerique Matricielle Et Optimisation-[Barra &Al]

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