Post on 22-Jul-2020
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimizat ion
Convex Optimization
Instructor: M�Dep. of CS & T
Tsinghua University
May 10, 2012
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimizat ion
Outline
1 Convex Optimization
2 TCP Duality
3 Rethinking Internet Traffic Management
4 SPEF: Optimal TE with OSPF
5 P2P Traffic Optimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Outline
1 Convex Optimization
2 TCP Duality
3 Rethinking Internet Traffic Management
4 SPEF: Optimal TE with OSPF
5 P2P Traffic Optimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Mathematical optimization
(mathematical) optimization problem
minimize f0(x)subject to fi(x) ≤ bi , i = 1, ...,m
(1)
x = (x1, ..., xn) : optimization variables
f0 : Rn → R : objective function
fi : Rn → R, i = 1, ...,m : constraint functions
optimal solution x⋆ has smallest value of f0 among all vectorsthat satisfy the constraints
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Solving optimization problems
general optimization problem
very difficult to solve
methods involve some compromise, e.g., very longcomputation time, or not always finding the solution
exceptions:certain problem classes can be solved efficiently andreliably
least-squares problems
linear programming problems
convex optimization problems
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Least-squares
minimize ‖Ax − b‖22 =k∑
i=1(aT
i x − bi)2 (2)
solving least-squares problems
analytical solution: x⋆ = (AT A)−1AT b
reliable and efficient algorithms and software
computation time proportional to n2k(A ∈ Rk×n);less ifstructured
a mature technology
using least-squares
least-squares problems are easy to recognize
a few standard techniques increase flexibility (e.g., includingweights,adding regularization terms)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Linear programming
minimize cT xsubject to aT
i x ≤ bi , i = 1, ...,m(3)
solving linear programsno analytical formula for solutionreliable and efficient algorithms and softwarecomputation time proportional to n2m if m ≥ n; less withstructurea mature technology
using linear programmingnot as easy to recognize as least-squares problemsa few standard tricks used to convert problems into linearprograms (e.g., problems involving ℓ1− or ℓ∞− norms,piecewise-linear functions)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Convex optimization problem
minimize f0(x)subject to fi(x) ≤ bi , i = 1, ...,m
(4)
objective and constraint functions are convex:
fi(αx + βy) ≤ αfi(x) + βfi(y)
if α+ β = 1, α ≥ 0, β ≥ 0
includes least-squares problems and linear programs asspecial cases
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
solving convex optimization problems
no analytical solution
reliable and efficient algorithms
computation time (roughly) proportional tomax{n3, n2m,F},where F is cost of evaluating fi ’s and theirfirst and second derivatives
almost a technology
using convex optimization
often difficult to recognize
many tricks for transforming problems into convex form
surprisingly many problems can be solved via convexoptimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Example
m lamps illuminating n (small, flat) patches
intensity Ik at patch k depends linearly on lamp powers pj:
Ik =m∑
j=1akjpj , akj = r−2
kj max{cos θkj , 0}
problem : achieve desired illumination Ides with bounded lamp powers
minimize maxk=1,...,n| lg Ik − lg Ides |subject to 0 ≤ pj ≤ pmax , j = 1, ...,m
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
how to solve?
use uniform power: pj = p, vary p
use least-squares:
Minn∑
k=1(Ik − Ides)
2
round pj if pj > pmax or pj < 0
use weighted least-squares:
Minn∑
k=1(Ik − Ides)
2 +m∑
j=1wj(pj − pmax/2)2
iteratively adjust weights wj until 0 ≤ pj ≤ pmax
use linear programming:
minimize maxk=1,...,n|Ik − Ides |subject to 0 ≤ pj ≤ pmax , j = 1, ...,m
which can be solved via linear programming
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
how to solve?(Cont.)
5. use convex optimization: problem is equivalent to
minimize f0(p) = maxk=1,...,n h(Ik/Ides)subject to 0 ≤ pj ≤ pmax , j = 1, . . . ,m
with h(u) = max{u, 1/u}
f0 is convex because maximum of convex functions is convexexact solution obtained with effort ≈ modest factor × least-squareseffort
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
additional constraints:does adding 1 or 2 below complicate the problem?
1. no more than half of total power is in any 10 lamps2. no more than half of the lamps are on (pj > 0)
answer: with (1), still easy to solve; with (2), extremely difficult
moral: (untrained) intuition doesn’t always work; without theproper background very easy problems can appear quitesimilar to very difficult problems
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Nonlinear optimization
traditional techniques for general nonconvex problems involvecompromiseslocal optimization methods (nonlinear programming)
find a point that minimizes f0 among feasible points near it
fast, can handle large problems
require initial guess
provide no information about distance to (global) optimum
global optimization methods
find the (global) solution
worst-case complexity grows exponentially with problem size
these algorithms are often based on solving convex subproblems
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Brief history of convex optimization
theory (convex analysis): ca1900õ1970algorithms
1939: the mathematical technique now known as linearprogramming(Leonid Kantorovich)
1947: simplex algorithm for liner programming(Dantzig)Leonid Vitaliyevich Kantorovich(1912-1986) was a Soviet/Russianmathematician and economist.He came up (1939) with the mathematicaltechnique now known as linearprogramming, some years before it wasreinvented and much advanced by GeorgeDantzig.He was the winner of the NobelPrize in Economics in 1975.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Brief history of convex optimization (Cont.)
theory (convex analysis): ca1900õ1970algorithms
1939: the mathematical technique now known as linearprogramming(Leonid Kantorovich)1947: simplex algorithm for liner programming(Dantzig)1960s: early interior-point methods(Fiacco&McCormick)1970s: ellipsoid method and other subgradient methods
In 1947 George Dantzig formulatedthe linear programming problem asa mathematical model for theplanning problem and devised thesimplex method for its solution,whichled to his titles as the/father oflinear programming0and the/inventor of the simplex method.0
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Brief history of convex optimization (Cont.)
algorithms
1980s: polynomial-time interior-point methods for linearprogramming (Karmarkar 1984)
late 1980sõnow: polynomial-time interior-point methods fornonlinear convex optimization (Nesterov&Nemirovski 1994)
applications
before 1990: mostly in operations research; few inengineering
since 1990: many new applications in engineering (control,signal processing, communications, circuit design,...)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Affine set
line through x1, x2: all points
x = θx1 + (1 − θ)x2
affine set: contains the line through any two distinct points in thesetexample: solution set of linear equations {x |Ax = b}(conversely, every affine set can be expressed as solution set ofsystem of linear equations)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Convex set
line segment between x1, x2: all points
x = θx1 + (1 − θ)x2
with 0 ≤ θ ≤ 1
convex set: contains line segment between any two points
x1, x2 ∈ C, 0 ≤ θ ≤ 1⇒ θx1 + (1 − θ)x2 ∈ C
examples(one convex, two nonconvex sets)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Convex combination and convex hull
convex combination of x1, ..., xk : any point x of the form
x = θ1x1 + θ2x2 + · · ·+ θk xk
with θ1 + θ2 + · · ·+ θk = 1, θi ≥ 0
convex hull conv S: set of all convex combinations of points in S
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Convex cone
conic (nonnegative) combination of x1 and x2: any point of the form
x = θ1x1 + θ2x2
with θ1 ≥ 0, θ2 ≥ 0
convex cone: set that contains all conic combinations of points inthe set
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Hyperplanes and halfspaces
hyperplane: set of the form {x |aT x = b}(a , 0)
halfspace: set of the form {x |aT x ≤ b}(a , 0)
a is the normal vector
hyperplanes are affine and convex; halfspaces are convex
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Norm balls and norm cones
norm : a function ‖ · ‖ that satisfies
‖x‖ ≥ 0; ‖x‖ = 0 if and only if x = 0
‖tx‖ = |t |‖x‖ for t ∈ R
‖x + y‖ ≤ ‖x‖+ ‖y‖
notation: ‖ · ‖ is general (unspecified) norm; ‖ · ‖symb is particular normnorm ball with center xc and radius r : {x |‖x − xc‖ ≤ r}
norm cone :{(x , t)|‖x‖ ≤ t}Euclidean norm cone is calledsecondorder conenorm balls and cones are convex
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Polyhedra
solution set of finitely many linear inequalities and equalities
ax � b ,Cx = d
(A ∈ Rm×n,C ∈ Rp×n ,� is componentwise inequality)
polyhedron is intersection of finite number of halfspaces andhyperplanes
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Separating hyperplane theorem
if C and D are disjoint convex sets, then there exists a , 0, bsuch that
aT x ≤ b for x ∈ C, aT x ≥ b for x ∈ D
the hyperplane {x |aTx = b} separates C and Dstrict separation requires additional assumptions (e.g., C is closed,D is a singleton)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Supporting hyperplane theorem
supporting hyperplane to set C at boundary point x0:
{x |aT x = aT x0}
where a , 0 and aT x ≤ aT x0 for all x ∈ C
supporting hyperplane theorem:
if C is convex, then there exists a supporting hyperplane at everyboundary point of C
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Definition
f : Rn → R is convex if dom f is a convex set and
f(θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y)
for all x , y ∈ dom f , 0 ≤ θ ≤ 1
f is concave if −f is convexf is strictly convex if dom f is convex and
f(θx + (1 − θ)y) < θf(x) + (1 − θ)f(y)for x , y ∈ dom f , x , y , 0 < θ < 1
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Examples on R
convex:
affine: ax + b on R, for any a, b ∈ R
exponential: eax , for any a ∈ R
powers: xa on R++, for a ≥ 1 or a ≤ 0
powers of absolute value: |x |p on R, for p ≥ 1
negative entropy: x log x on R++
concave:
affine: ax + b on R, for any a, b ∈ R
powers: xa on R++, for 0 ≤ a ≤ 1
logarithm: log x on R++
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
First-order condition
f is differentiable if dom f is open and the gradient:
∇f(x) = (∂f(x)∂x1
,∂f(x)∂x2
, · · · , ∂f(x)∂xn
)
exists at each x ∈ dom f
1st-order condition:
differentiable f with convexdomain is convex iff
f(y) ≥ f(x) + ∇f(x)T (y − x) forall x , y ∈ dom f
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Second-order conditions
f is twice differentiable if dom f is open and the Hessian∇2f(x) ∈ Sn:
∇2f(x)ij =∂2f(x)∂xi ∂xj
, i, j = 1, . . . , n
exists at each x ∈ dom f
2nd-order condition:for twice differentiable f with convex domain
f is convex if and only if
∇2f(x) � 0 for all x ∈ dom f
if ∇2f(x) ≻ 0 for all x ∈ dom f , then f is strictly convex
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Examples
1. quadratic function: f(x) = (1/2)xT Px + qT x + r (with P ∈S)
∇f(x) = Px + q, ∇2f(x) = P
convex if P � 02. least-squares objective: f(x) = ‖Ax − b‖22
∇f(x) = 2AT(Ax − b), ∇2f(x) = 2AT A
convex (for any A)3. quadratic-over-linear : f(x , y) = x2/y
∇2f(x , y) = 2y3
[
y−x
] [
y−x
]T
� 0
convex for y > 0
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Optimization problem in standard form
minimize f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m; hi(x) = 0, i = 1, . . . , p
x ∈ Rn is the optimization variablef0 : Rn → R is the objective or cost functionfi : Rn → R, i = 1, . . . ,m, are the inequality constraintfunctionshi : Rn → R are the equality constraint functions
optimal value:
p∗ = inf {f0(x)|fi(x) ≤ 0, i = 1, . . . ,m, hi(x) = 0, i = 1, . . . , p}p∗ = ∞ if problem is infeasible (no x satisfies the constraints)
p∗ = −∞ if problem is unbounded below
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Optimal and locally optimal points
x is feasible if x ∈ dom f0 and it satisfies the constraintsa feasible x is optimal if f0(x) = p∗;Xopt is the set of optimal points
x is locally optimal if there is an R > 0 such that x is optimal for
minimize(overz) f0(z)subject to fi(z) ≤ 0, i = 1, . . . ,m, hi(z) = 0, i = 1, . . . , p
‖z − x‖2 = [(z1 − x1)2 + · · ·+ (zn − xn)
2]1/2 ≤ R
examples (with n = 1,m = p = 0)
f0(x) = 1/x, dom f0 = R++ : p∗ = 0, no optimal point
f0(x) = − log x, dom f0 = R++ : p∗ = −∞f0(x) = x log x, dom f0 = R++ : p∗ = −1/e, x = 1/e, is optimal
f0(x) = x3 − 3x, p∗ = −∞, local optimum at x = 1
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Implicit constraints
the standard form optimization problem has an implicit constraint
x ∈ D =⋂m
i=0dom fi ∩⋂p
i=1dom hi
we call D the domain of the problem
the constraints fi(x) ≤ 0, hi(x) = 0 are the explicit constraints
a problem is unconstrained if it has no explicit constraints(m = p = 0)
example
minimize f0(x) = −k∑
i=1log(bi − aT
i x)
is an unconstrained problem with implicit constraints aTi x < bi
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Feasibility problem
find xsubject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p
can be considered a special case of the general problem withf0(x) = 0:
minimize 0subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p
p∗ = 0 if constraints are feasible; any feasible x is optimal
p∗ = ∞ if constraints are infeasible
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Convex optimization problem
example
minimize f0(x) = x21 + x2
2subject to f1(x) = x1/(1 + x2
2 ) ≤ 0h1(x) = (x1 + x2)
2 = 0
f0 is convex; feasible set {(x1, x2)|x1 = −x2 ≤ 0} is convex
not a convex problem (according to our definition): f1 is not convex,h1 is not affine
equivalent (but not identical) to the convex problem
minimize x21 + x2
2subject to x1 ≤ 0
x1 + x2 = 0
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Local and global optimal
any locally optimal point of a convex problem is (globally) optimal
proof:suppose x is locally optimal and y is optimal with f0(y) < f0(x)x locally optimal means there is an R > 0 such that
z feasible, ‖z − x‖2 ≤ R ⇒ f0(z) ≥ f0(x)
consider z = θy + (1 − θ)x with θ = R/(2‖y − x‖2)
‖y − x‖2 > R, so 0 < θ < 1/2z is a convex combination of two feasible points, hence also feasible
‖z − x‖2 < R when θ is small enough and
f0(z) ≤ θf0(x) + (1 − θ)f0(y) < f0(x)
which contradicts our assumption that x is locally optimal
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Optimality criterion for differentiable f0
x is optimal if and only if it is feasible and
∇f0(x)T (y − x) ≥ 0 for all feasible y
if nonzero, ∇f0(x) defines a supporting hyperplane to feasible set Xat x
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
unconstrained problem : x is optimal if and only if
x ∈ dom f0, ∇f0(x) = 0
equality constrained problem
minimize f0(x) subject to Ax = b
x is optimal if and only if there exists a ν such that
x ∈ dom f0, Ax = b , ∇f0(x) + ATν = 0
minimization over nonnegative orthant
minimize f0(x) subject to x � 0
x is optimal if and only if
x ∈ dom f0, x � 0, { ∇f0(x)i ≥ 0 xi = 0∇f0(x)i = 0 xi > 0
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Lagrangian
standard form problem (not necessarily convex)
minimize f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p
variable x ∈ Rn, domain D, optimal value P∗
Lagrangian:L : Rn × Rm × Rp → R, with dom L = D × Rm × Rp ,
L(x , λ, ν) = f0(x) +m∑
i=1λi fi(x) +
p∑
i=1νihi(x)
weighted sum of objective and constraint functions
λi is Lagrange multiplier associated with fi(x) ≤ 0
νi is Lagrange multiplier associated with hi(x) = 0Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Lagrange dual function
Lagrange dual function: g = Rm × Rp → R
g(λ, ν) = infx∈DL(x , λ, ν)
= infx∈D(f0(x) +m∑
i=1λi fi(x) +
p∑
i=1νihi(x))
g is concave, can be −∞ for some λ, ν
lower bound property: if λ � 0, then g(λ, ν) ≤ p∗
Proof
if x̃ is feasible and λ � 0, then
f0(x̃) ≥ L(x̃ , λ, ν) ≥ infx∈DL(x , λ, ν) = g(λ, ν)
minimizing over all feasible x̃ gives p∗ ≥ g(λ, ν)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Primal & Dual problem
Primal problem
minimize f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p
variable x ∈ Rn, domain D, optimal value P∗
Dual problem
maximize g(λ, ν)subject to λ � 0
g(λ, ν) = infx∈D(f0(x) +m∑
i=1λi fi(x) +
p∑
i=1νihi(x))
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimizationéó¯K�¢~¯K£ã,7ì�m\ó�÷7�Â\Ǒ56�/÷!\ó7IÂ\Ǒ30�/÷.\ó�÷7�I7ó4��§hÖó2��¶\ó�÷7II7ó3��§hÖó1��.T�mzF�^�7óÚhÖóoó�©OǑ120Ú50��.TXÛSSSüüü)))���âU�zFÂ\��ºÂÂÂ\\\������zzz�.maximize z = 56x1 + 30x2;subject to 4x1 + 3x2 ≤ 120,
2x1 + x2 ≤ 50,x1, x2 ≥ 0.
x∗ = (x1, x2)T = (15, 20)T , z∗ = 1440.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimizationéó¯K�¢~l,��ÆÝ�ÄT¯Keyk���N²Eö§�¥k�17ì�¬£7�Ú7I¤)�¾ü.��|^T7ì�m�7óÚhÖó5�¤�¥�¾ü. �o§�I�ïÄXÛ�7óÚhÖó�óóó���½½½ddd§âUQ�T�mú�k|�ãl �¿O�\ó§q�gC¤G�ó�¤^o���.¤¤¤^̂̂������zzz�.minimize f = 120ω1 + 50ω2;subject to 4ω1 + 2ω2 ≥ 56,
3ω1 + ω2 ≥ 30,ω1, ω2 ≥ 0.
ω∗1 = 2, ω∗2 = 24, f∗ = 1440.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Duality theorem
Weak duality theorem
Let x′ be a feasible solution to the primal problem, and λ′ be afeasible solution to its dual problem, then f(x′) ≥ g(λ′)..
Strong duality theorem (for convex programming)
Assume that X is an open convex set, f0(x) and fi(x)(i = 1, . . . ,m)are convex functions, and hi(x)(i = 1, . . . , p) are linear functions.Moreover, assume that there exists x′ ∈ X suchfi(x′) < 0, (i = 1, . . . ,m) and hi(x′) = 0, (i = 1, . . . , p), then
i) The optimal value of the primal problem is equal to that of thedual problem.
ii) Let x∗ be a solution of the primal problem and λ∗ be a solutionof the dual problem, then it holds that λ∗i fi(x∗) = 0, (i = 1, . . . ,m).
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Karush-Kuhn-Tucker (KKT) conditions
The following four conditions are called KKT conditions(for a problem with differentiable fi , hi)1. primal constraints: fi(x) ≤ 0, i = 1, . . . ,m, hi(x) = 0, i = 1, . . . , p2. dual constraints: λ � 03. complementary slackness: λi fi(x) = 0, i = 1, . . . ,m4. gradient of Lagrangian with respect to x vanishes:
∇f0(x) +m∑
i=1λi∇fi(x) +
p∑
i=1νi∇hi(x) = 0
if strong duality holds and x , λ, ν are optimal, then they must satisfythe KKT conditions
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
Slater’s condition
Convex Optimization Problem
minimize f0(x)subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p
variable x ∈ Rn, domain D.
We say that the problem satisfies Slater’s condition if it is strictlyfeasible, that is
∃x0 ∈ D : fi(x0) < 0, i = 1, . . . ,m, hi(x0) = 0, i = 1, . . . , p.
Slater’s condition (or Slater condition) is a sufficient condition forstrong duality to hold for a convex optimization problem.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization
KKT conditions for convex problem
if x̃ , λ̃, ν̃ satisfy KKT for a convex problem, then they are optimal:
from complementary slackness: f0(x̃) = L(x̃ , λ̃, ν̃)
from 4th condition (and convexity): g(λ̃, ν̃) = L(x̃ , λ̃, ν̃)hence, f0(x̃) = g(λ̃, ν̃)
If Slater’s condition is satisfied:x is optimal if and only if there exist λ, ν that satisfy KKT conditions
recall that Slater implies strong duality, and dual optimum isattained
generalizes optimality condition ∇f0(x) = 0 for unconstrainedproblem
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization�{�^�ê��`z�{��eü{Úî{�ÝFÝ{¨v¼ê{:v¼ê{S:v¼ê{�f{Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization�^�ê��`z�{��eü{minimize f(x)subject to x ∈ En
f(x)äk��ëY �ê,�|¢��µd(k ) = −∇f(x(k ))Úî{Ä�g�:^���g¼ê�Cq8I¼êf(x),,�°(/�Ñù��g¼ê�4�:.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Op timal TE with OSPF P2P Traffic Optimization¨v¼ê{:v¼ê{: Ú\v�p(x) =
m∑
i=1ϕ[gi(x)] +
l∑
j=1ψ[hj(x)]
where ϕ[gi(x)] = [max{0,−gi(x)}]α ψ[hj(x)] = |hj(x)|βS:v¼ê{: æN¼êG(x , r) = f(x) + rB(x)
where : B(x) =m∑
i=1
1gi(x)
B(x) = −m∑
i=1lngi(x)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Outline
1 Convex Optimization
2 TCP Duality
3 Rethinking Internet Traffic Management
4 SPEF: Optimal TE with OSPF
5 P2P Traffic Optimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
TCP & AQM
Duality→ equilibrium
Source rates xi(t) are primal variables
Congestion measures pl(t) are dual variables
Congestion control is optimization process
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Congestion control
Interaction of source rates xs(t) and congestion measures pl(t)Duality theory
They are primal and dual variables
Flow control is optimization process
Example congestion measure
Loss (Reno)
Queueing delay (Vegas)
Queue length (RED)
Price (REM)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Model
Sources s
L(s) - links used by source s
Us(xs) - utility if source rate = xs
Network
Links l of capacities cl
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Primal problem
maximizexs≥0∑
sUs(xs)
subject to x l ≤ cl , ∀l ∈ L
Assumptions: Strictly concave increasing Us
Unique optimal rates xs exist
Direct solution impractical
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Duality Approach
Primal
maximizexs≥0∑
sUs(xs)
subject to x l ≤ cl , ∀l ∈ L
Dual
minimizep≥0 D(p) = (maximizexs≥0∑
sUs(xs) +
∑
lpl(cl − x l))
Primal-dual algorithm:
x(t + 1) = F(p(t), x(t))
p(t + 1) = G(p(t), x(t))
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Duality Model of TCP
Source algorithm iterates on rates
Link algorithm iterates on prices
With different utility functions
Primal-dual algorithm:
x(t + 1) = F(p(t), x(t)) ⇐ Reno,Vegasp(t + 1) = G(p(t), x(t)) ⇐ DropTail,RED,REM
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Summary
Flow control problem
maximizexs≥0∑
sUs(xs)
subject to x l ≤ cl , ∀l ∈ L
Primal-dual algorithm:
x(t + 1) = F(p(t), x(t))p(t + 1) = G(p(t), x(t))
Major TCP schemes
Maximize aggregate source utility
With different utility functions
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Summary
What are the (F ,G,U)?Derivation
Derive (F ,G) from protocol description
Fix point (x, p) = (F ,G) gives equilibrium
Derive U: regard fixed point as Kuhn-Tucker condition
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Active queue management
Idea: provide congestion information by probabilistically markingpacketsIssues
How to measure congestion (pandG)?
How to embed congestion measure?
How to feed back congestion info?
x(t + 1) = F(p(t), x(t)) ⇐ Reno,Vegasp(t + 1) = G(p(t), x(t)) ⇐ DropTail,RED,REM
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
RED (Floyd & Jacobson 1993)
Congestion measure: average queue length
bl(t + 1) = [bl(t) + y l(t) − cl ]+
rl(t + 1) = (l − a)rl(t) + a ∗ bl(t)
Embedding: p-linear probability function
Feedback: dropping or ECN markingInstructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
REM (Athuraliya & Low 2000)
Congestion measure: price
rl(t + 1) = [rl(t) + γ(αlbl(t) + y l(t) − cl)]+
Embedding: exponential probability function
Feedback: dropping or ECN markingInstructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Key features
Clear buffer and match rate
rl(t + 1) = [rl(t)+ γ(αlbl(t)+ y l(t) − cl)]+
Clearbuffer matchrate
Sum prices
pl(t) = 1 − φ−rl(t)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
TCP and AQM
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Reno:F
for every ack (ca){ w+ = 1
w }
for every loss{ w := w
2 }
∆ws(t) =xs(t)(1−q(t))
ws− xs(t)q(t) ∗ 1
2 ∗4ws
3
xs(t) = ws(t)/Ds ,Ds is the equilibrium round trip time
Fs(q(t), x(t)) = xs(t) +1−q(t)
D2s− 2x2
s (t)3 q(t)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE wit h OSPF P2P Traffic Optimization
Reno: Utility Function
Fs(q(t), x(t)) = xs(t) +1−q(t)
D2s− 2x2
s (t)3 q(t)
xs = Fs(q, x)
⇒ 1−q(t)D2
s=
2x2s
3 q
⇒ 33+2x2
s D2s= q ⇒ Ureno
s (xs) =√
3/2Ds
tan−1(√
23xsDs)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Outline
1 Convex Optimization
2 TCP Duality
3 Rethinking Internet Traffic Management
4 SPEF: Optimal TE with OSPF
5 P2P Traffic Optimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Traffic Management Today
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Shortcomings of today.s traffic management
Congestion control assumes routing is fixed, trafficengineering assumes traffic is inelastic
Link weight tuning problem is NP-hard
Link weights are tuned at the timescale of hours, slower thantraffic shifts
What if we redesign traffic management from scratch usingoptimization tools?
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Congestion Control Implicitly Maximizes AggregateUser Utility
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Traffic Engineering Explicitly Minimizes NetworkCongestion
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
A Balanced Objective
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffic Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Model of He et al and TRUMP Algorithm
maximize∑
s Us(xs) − w∑
l f(zl/cl)subject to x = Hy, z = Ay
y ≥ 0, z ≤ c,
f is a convex, non-decreasing,and twice-differentiablefunction , e.g., ezl/cl .
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Outline
1 Convex Optimization
2 TCP Duality
3 Rethinking Internet Traffic Management
4 SPEF: Optimal TE with OSPF
5 P2P Traffic Optimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Motivation
Traffic Engineering (TE) is important for ISPs
OSPF is a commonly used intra-domain routing protocol
The quality of OSPF-based traffic engineering dependslargely on the choice of link weights
Optimizing the link weights for OSPF to the offered traffic is anknown NP-hard problem
Some related issues of multi-path protocol are addressed in ourprevious work LBMP
Introduce a novel logarithm barrier-based approach to makesoptimal solution achievable
Demonstrate a distributed implementation and design apractical Logarithm-Barrier-based Multi-path Protocol (LBMP)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Challenges
Can we guarantee the universal existence of optimal link weights?
Wang et al. (INFOCOM’01) proved that any arbitrary set ofroutes can be converted to shortest paths with respect tosome set of positive link weightsWe prefer to ensure the universal existence of optimal linkweights which are explicitly determined by the objectivefunction
Difficulties in OSPF protocol to realize/optimal routing0It uses shortest path routing with destination based forwardingThe forwarding mechanism equally splits the correspondingtraffic across the multiple equal cost paths or next hops for agiven destination routing prefix.
Open problem: can a link-state protocol with hop-by-hopforwarding achieve optimal TE?
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Related Work
Traffic Engineering
Fortz et al. showed that optimizing the link weights for OSPF withevenly split over ECMP to the offered traffic is an NP-hard problem
Xu et al. (INFOCOM’08) proposed PEFT, a link-state routingprotocol splitting traffic over multiple paths with an exponentialpenalty on longer paths. Limitations:- All available paths rather than the shortest paths are involved- No illustration on how to find the candidate paths for traffic splitting
Rate Control and Congestion avoidance
Kelly et al. considered fairness and stability from theperspective of economic and engineering
maximize∑
s Us(xs)subject to Ax ≤ c
x ≥ 0Us(xs) =
{
ps log(xs), α = 1ps(1 − α)−1x1−α
s , otherwise.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Main Contributions
Ensuring the existence of optimal link weights, which can beexplicitly formulated using the optimal distribution of traffic andobjective function
Developing a new link-state routing protocol, SPEF, andtheoretically proving that it can achieve the optimal TE forintra-domain IP networks
Illustrating the effectiveness of SPEF in terms of link utilizationand network traffic distribution via simulations
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Network Model
Directed network G = (N ,E) with vertex set N , edge set E,the capacity for link (i, j) ∈ E is cij .Source-destination pairs set R = {(s1, t1), · · · , (sR , tR)}, thedemand for (sr , tr) is dr .Destination node set with D = {t ∈ N : ∃ r ∈ R s.t. tr = t}.The flow of commodity t along edge (i, j) is f t
ij .Two kinds of constraints:Capacity constraints:
fij =∑
t∈Df tij ≤ cij ,∀(i, j) ∈ E
Flow conservation constraints:∑
i:(s,i)∈Ef tsi −
∑
j:(j,s)∈Ef tjs = dt
s , ∀t ∈ D, s ∈ N
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Universal Existence of Optimal Link Weights
TE(V ,G, c,D)
maximizeft≥0∑
(i,j)∈E Vij(sij)
subject to c −∑t∈D ft = s ≥ 0Bf t = dt , ∀t ∈ D,
Let (s, ft , t ∈ D) be a solution and w = (wij : (i, j) ∈ E) be thecorresponding Lagrangian multiplier vector.
Result 1 We have that ft determines the shortest path for eachsource-destination pair (s, t) under the link weight vector w, whichis determined explicitly by the utility function Vij and the residualcapacity sij as wij = V ′ij(sij) if link (i, j) ∈ E is no saturated, i.e.sij > 0.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Properties of Optimal Link Weights
Linkij(Vij ;wij):maximize Vij(sij) − wijsij
subject to sij ≥ 0.
Network(G, c,D;w)
maximize∑
(i,j)∈E wijsij
subject to c −∑
t∈D ft = s ≥ 0
Bf t = dt , ∀t ∈ Dft ≥ 0.
minimize∑
(i,j)∈E wij∑
t∈D f tij
subject to∑
t∈D f tij ≤ cij ,∀(i, j) ∈ E
Bf t = dt , ∀t ∈ Dft ≥ 0.
Result 2 There exists a weight vector w = (wij , (i, j) ∈ E) such thatthe vector s = (sij , (i, j) ∈ E), formed from the unique solution sij toLinkij(Vij ;wij), solves Network(G, c,D;w). The vector s also solvesTE(V ,G, c,D).
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
(q, β) Utility Function
Consider the objective function
Vij(sij) =
qij log (sij), if β = 1qij(1 − β)−1cβ−1
ij s1−βij , if β , 1.
Example 1 For β = 2, let qij = cij .
Vij(sij) = −1 − fij/(cij − fij), wij = cij/(cij − fij)2.
Example 2 For β = 1, let qij = 1.
Vij(sij) = log (sij), wij = 1/(cij − fij).
Example 3 For β = 0, let dij be the processing and propagation delay onlink (i, j).
Vij(sij) = dijcij − dij fij , wij = dij .
Remark : wij = dij for unsaturated link (i, j) and wij ≥ dij for saturated link(i, j). If dij = 1, we have the minimum hop routing.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
SPEF: Toward the Optimal Traffic Engineering with OSPF
In SPEF (Shortest paths Penalizing ExponentialFlow-splitting), we only need one more weight for each link .
The packet forwarding scheme is the same as that in OSPF:hop-by-hop along the shortest paths constructed based ondestination according to the first link weights.
When there are multiple equal cost shortest paths for someOD pairs in view of the first link weights, the flow split ratioover the multiple shortest paths can be independentlycomputed by the routers from the second link weights.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Shortest-Path-Based NEM
We can determine the set of shortest paths ON = {ONt : t ∈ D}based on the first link weights, where ONt is the shortest path setfor any node s ∈ N to destination t ∈ D. Specifically, SPr denotesthe shortest path set for (sr , tr). Let SP = {SPr : r ∈ R}.We use NEM (Network Entropy Maximization) to deal with thescenario of multiple equal cost paths.NEM(SP, f∗,D):
maximize −∑
r∈R dr∑nr
k=1 prk log pr
ksubject to
∑
r∈R∑
k :(i,j)∈SPrk
drprk ≤ f∗ij ,∀(i, j) ∈ J
∑nrk=1 pr
k = 1, ∀r ∈ R,
where SPrk denotes the k -th shortest path from sr to tr and pr
k isthe traffic split ratio for path SPr
k .
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Forwarding Table for SPEF Routing
Table 2 Forwarding table for SPEF routing.
Lengths of multiple equal cost shortest paths through
Next hop link (s, next hop) to t in view of the second link weights
v1 (v(s,t)11 , · · · , v(s,t)
1n1)
......
vms (v(s,t)ms1 , · · · , v
(s,t)msnms
)
The traffic to destination t can be split according to the followingformula
Γt(s, vk ) =
∑nkj=1 e−v(s,t)
kj
∑msi=1
∑nij=1 e−v(s,t)
ij
, k = 1, · · · ,ms. (5)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
An Example
96
10 123
1 2
78
1
2 3
4
5 6
7
5
11
13
4There are 4 source-destination pairs and each needs a bandwidthof 4 units. Each link has a capacity of 5 units.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
An Example (Continues)
1 2 3 4 5 6 7 8 9 10 11 12 130
0.5
1
1.5
2
2.5
3
Links
The
Firs
t Link
Weig
hts
SPEF0
SPEF1
SPEF5
1 2 3 4 5 6 7 8 9 10 11 12 130
0.5
1
1.5
2
2.5
Links
The
Seco
nd L
ink W
eight
s
SPEF0
SPEF1
SPEF5
1 2 3 4 5 6 7 8 9 10 11 12 130
0.5
1
1.5
2
Links
Link U
tiliza
tions
OSPFSPEF
0
SPEF1
SPEF5
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Outline
1 Convex Optimization
2 TCP Duality
3 Rethinking Internet Traffic Management
4 SPEF: Optimal TE with OSPF
5 P2P Traffic Optimization
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P6þ`z�.`zg´ÏLÿþ&EÚÝ�.��P2P6þÝÏLP2P6þÝ�)ó´þP2P6þ©ÛP2P`zEâÜÝ�Yéó´P2P6þ�KǑÏL�`zï�©Û�)�`�ÜÝ�YInstructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P6þ`zÏLP2P6þÝ��ó´þ�P2P6þÏLP2P6þÝÚ´d&E§�±���äó´þ�P2P6þò�ä�ó´8ܽÂǑJ§Kà�à�´dr �±½ÂǑJ���f8§ǑÒ´�|ó´�8ܽÂ0-1ÝA : �ó´jáu´di�,Aij = 1;ÄKAij = 0�^ó´þ�6þÒ´:L = Ay
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P`zEâéP2P6þÝ�KǑP2P6þ�/z
uij = f(dij),
{
f(dij) ≥ 1, dij ≤ Sf(dij) < 1, dij > S
P2P6þ��üÑuij =
(1 − λij)uij , i , juij +
∑
i,jλuij , i = j , Pj ⊆ CacheSet
uij , Pj & CacheSet
P2P6þ½�Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
The basic modelüó´Â�ï�KǑÏ�-`z~��6þ;ó´�§Ý;ó´|^Çó´Â�¼ê
Bi =∫ νi
0ψi(ηi)dν =
∫ νi
0ψi(
ui−νCi
)dν�äÂ�ï��ä�ó´Â�\�Bt =
m∑
i=1Bi =
m∑
i=1
∫ νi
0ψi(
ui−νCi
)dν
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P6þ`z�`zï�maximize B(xi)subject to
∑
xi∈Xi
ci ≤ c�þXi(< x1, x2, ..., xn >)L«P2P`zEâ���ÜÝ�YciL«ÜÝ�YXi�ÜÝm�§ cK´��m�þ�B(Xi)L«3ÜÝ�YXie�äÜÝoÂ�3ÜÝm���åe§�)�äÜÝoÂ����ÜÝ�Y.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P6þ`zP2P6þ��-yk���ó�Ì�8¥3üó´���{�O-"yéu��ÜÝ�©Û�Û���ÜÝ�Y-��KǑ�ó´U§�JØU{üU\-ÀJ�Pl��ó´ÜÝ¿Ø´�`üÑ
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimizationõó´��ÜÝ��`z¯K`z�.maximize Btotal =
m∑
i=1xiBi(X ∗i )
subject tom∑
i=1xici ≤ c
xi = 0or1õó´��ÜÝ�E,Ý-�5�Ǒ0/1��¯K-õ�ª�mØ�)õó´��ÜÝ�{-©|½.�{£�`�{¤-éuª�{
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P6þ`zµd'�n��{��J-{ü�À��Pló´��{5U��-éuª�{�Cu�`�{5Uéuª�{�J��O�m��
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
P2P6þ`zµd�Pl�20^ó´���Uõ�¹ÜÝ�.Ú�{�±k��Uõó´�Pl
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Future Research Directions
Nonconvexity
- dealing with nonconvex optimization formulations
Research Issues Involving Time
- dealing with formulations of real-time applications
Quantifying Network X-ities
- e.g., scalability, availability, diagnosability, manageability, etc.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Nonconvexity
Four different forms of nonconvex optimization formulations
Nonconcave utilitye.g., the sigmoidal utility that are more realistic models inapplications
Nonconvex constraint sete.g., lower bounds on SIR as a function of transmit powervector
Integer constrainte.g., formulations in single-path routing protocols
Convex constraints growing exponentially with the number ofvariablese.g., schedulability constraints in multi-hop interferencemodels
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Three approaches to deal with nonconvexity
Go around nonconvexity (Left figure)-Change variables to turn the seemingly nonconvex probleminto a convex one, e.g., geometric programming
Go through nonconvexity (Middle figure)-Use successive convex relaxations, utilize special structures,or leverage smarter branch and bound methods
Go above nonconvexity (Right figure)-Design for optimizability, changing assumptions may result ina different, much easier-to-solve formulation
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Geometric Programming
Original Form
minimize f0(x)subject to fi(x) ≤ 1, i = 1, ...,m
hi(x) = 1, i = 1, ..., p(6)
where fi() are posynomials and hi() are monomials.f(x) = cxa1
1 xa22 · · · x
ann , c > 0 and ai ∈ R
Define: yi = log(xi), then f(x) = eaT y+b , where b = log(c)
Convex Form
minimize f0(ey)
subject to fi(ey) ≤ 1, i = 1, ...,mhi(ey) = 1, i = 1, ..., p
(7)
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Research Issues Involving Time
Utility functions-For real-time applications, utility should be a function of rateallocation through the transients instead of equilibrium rates.-How to formulate and maximize such functions?
Variants in time scales-Time scales differ by several orders-of-magnitude acrosslayers. e.g., App., Trans. and Phys. are determined by userbehaviors, RTT and physics of the transmission medium.-How to bound them tightly?
From theoretical to practical-Theoretically, iterative algorithms allow variables drop belowa threshold during the transient, such as TCP window size-Practically, under threshold may result in disconnection.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Quantifying Network X-ities
Network X-ities: robustness metrics-Performance Metrics: throughput, latency, and distortion.-Robustness Metrics: evolvability, scalability, availability,diagnosability, and manageability.Existing techniques for ”design by decomposition”-Enhance scalability and evolvability.-Present risks to manageability such as diagnosability andoptimizability.long-term goal of Optimization Decomposition-Move away from the restriction of one decomposition fits all,away from the focus on deterministic fluid models andasymptotic convergence, and even away from the emphasison optimality-Instead, towards optimizability.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
References 1
S. Boyd, L. Vandenberghe. Convex Optimization, CambridgeUniversity Press: Cambridge,2004.
S. H. Low. A Duality Model of TCP and Queue ManagementAlgorithms S. H. Low, ”A Duality Model of TCP and QueueManagement Algorithms,” IEEE/ACM Trans. Netw. vol. 11, pp.525õ536, Aug. 2003.
J. He, M. Suchara, M. Bresler, J. Rexford, and M. Chiang. Frommultiple decompositions to TRUMP: traffic management usingmultipath protocol. Summitted to IEEE/ACM Trans. Networking,March 2008.�²�,Çï²,M�. Peer-to-Peer6þ��ÜÝï�"�u�ÆÆ�£g,�Æ�¤, 2009 1Ï.
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
References 2
S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. A tutorial ongeometric programming. Optimization and Engineering, vol. 8, no.1, pp. 67-127, 2007
K. Xu, H. Liu, J. Liu and M. Shen. One More Weight is Enough:Toward the Optimal Traffic Engineering with OSPF. In Proc. of IEEEICDCS, 2011
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization
Convex Optimization TCP Duality Rethinking Internet Traffi c Management SPEF: Optimal TE with OSPF P2P Traffic Optimization
Thanks for Your Attentions
Instructor: MMM��� OOO���ÅÅÅ���äää ÷÷÷ïïïÄÄÄConvex Optimization