Introduction to Convex Optimization

Chee Wei Tan

CS 8292 : Advanced Topics in Convex Optimization and its ApplicationsFall 2010

Outline

• Optimization

• Examples

• Overview of Syllabus

• An Example: The Illumination Problem

• References

Acknowledgement: Thanks to Mung Chiang (Princeton),

Stephen Boyd (Stanford) and Steven Low (Caltech) for the course

materials in this class.

1

Optimization Mentality

OPT

OPT

OPT

Theories

SolutionsProblems

Optimization as a language, as an engineering system, as a unifyingprinciple

2

Optimization

minimize f(x)subject to x ∈ C

Optimization variables: x. Constant parameters describe objectivefunction f and constraint set C

3

Questions

• How to describe the constraint set?

• Can the problem be solved globally and uniquely?

• What kind of properties does it have? How does it relate to

another optimization problem?

• Can we numerically solve it in an efficient, robust, and distributed

way?

• Can we optimize over a sequence of time instances?

• Can we find the problem for a given solution?

4

Solving Optimization Problems

• general nonlinear optimization problem

often difficult to solve, e.g., if nonconvexmethods involve some compromise, e.g., very long computational time

• local optimization

find a suboptimal solutioncomputationally fast but initial point dependent

• global optimization

find a global optimal solutioncomputationally slow

In convex optimization, the art and challenge is in problem formulation. Innonconvex optimization, the art and challenge is in problem structure. Convex

optimization plays an important role in exploiting this problem structure

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Perspective

Widely known: linear programming is powerful and easy to solve

Modified view: ... the great watershed in optimizationisn’t between linearity and nonlinearity, but

convexity and nonconvexity.– SIAM Review 1993, R.Rockafellar

• Local optimality is also global optimality

• Lagrange duality theory well developed

• Know a lot about the problem and solution structures

• Efficiently compute the solutions numerically

So we’ll start with convex optimization introduction, then specialize into differentapplications involving convex problems (and nonconvex problems)

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Examples

• Least-Squares:minimizex ‖Ax− b‖2

Analytical solution: x? = (ATA)−1AT b; reliable and efficient algorithms andsoftware; computational time proportional to n2k (A ∈ Rk×n); less withstructure; a mature technology

• Using Least-Squares: several standard techniques increase the flexibility of leastsquares, e.g., adding weights, regularization

• Linear Programming:

minimize cTxsubject to aTi x ≤ bi, i = 1, . . . ,m

No analytical formula for solution; reliable and efficient algorithms and software;computational time proportional to n2m if m ≥ n; less with structure; a mature

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technologyNot as easy to recognize as least squares

• Convex Optimization:

minimize f0(x)subject to fi(x) ≤ bi, i = 1, . . . ,m

No analytical formula for solution; reliable and efficient algorithms and software;computational time (roughly) proportional to max{n3, n2m,F} where F is thecost of evaluating the objective and constraint functions fi and their first andsecond order derivatives; almost a technology

• Using convex optimization

Often difficult to recognize, many tricks for transforming problems, surprisinglymany problems can be solved (or be tackled elegantly) via convex optimization

Once the formulation is done, solving the problem is, like least-squares or linearprogramming, (almost) technology. Learn by examples and applications

8

Why Applications?

Draw inferences about other cases from one instance

Need to know how to recognize and formulate convex optimization, and useoptimization tools to solve your problem (an objective of this course)

9

Convex Sets and Convex Functions

f(y)

(x, f(x))

f(x) + ∇f(x)T (y − x)

10

Convex Optimization and Lagrange Duality

11

Linear Programming and QuadraticProgramming

P

x∗

−c

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Geometric Programming

(a + b)/2(a + b)/2

√

ab

13

Semidefinite Programming

K

K∗

x

y

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Decomposition

Master Problem

Subproblem 1...

Secondary Master Problem

prices / resources

First LevelDecomposition

Second LevelDecomposition

Subproblem

Subproblem N

prices / resources

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Internet Congestion Control &Network Utility Maximization

Linux TCP Linux TCP FAST

Average

utilization

19%

27%

92%

txq= 100 txq= 10000

95%

16%

48%

Linux TCP Linux TCP FAST

2G

1G

txq= 10000txq= 100

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Power Control in Wireless

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Uncovering the Perron-Frobenius Root

0 1 2 3 4 50

1

2

3

4

5

r1

r 2

Rate region (units in nats/symbol)

w = x(F + (1/p̄i)veTi ) ◦ y(F + (1/p̄i)ve

Ti )

log(1 + 1/ρ(F + (1/p̄i)veTi ))(1, 1)T

90 degrees

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Combinatorial Problems

Maximize the number of edges between a subset and itscomplementary subset

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Convex-Cardinality Problems

Sparsity, Compressed Sensing, Single-Path Routing

20

Data Mining

Face ≈ eyes + nose + lip + chin + etc

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Illumination Problem

m lamps illuminating n (small, flat) patches

Intensity Ik at patch k depends linearly on lamp powers pj:

Ik =m∑j=1

akjpj, akj = r−2kj max{cos θkj, 0}

problem: achieve desired illumination Ides with bounded lamp powers

minimize maxk=1,...,n

| log Ik − log Ides|

subject to 0 ≤ pj ≤ pmax, j = 1, . . . ,m

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How to Solve Illumination Problem

1. use uniform power: pj = p, vary p

2. use least-squares:

minimizen∑k=1

(Ik − Ides)2

round pj if pj > pmax or pj < 0

3. use weighted least-squares:

minimizen∑k=1

(Ik − Ides)2 +

m∑j=1

wj (pj − pmax)2

iteratively adjust weights wj until 0 ≤ pj ≤ pmax

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4. use linear programming:

minimize maxk=1,...,n

|Ik − Ides|

subject to 0 ≤ pj ≤ pmax, j = 1, . . . ,m

which can be solved via linear programming

of course these are suboptimal ‘solutions’ (to the original problem)

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}

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0 1 2 3 40

1

2

3

4

5

u

h(u

)

f0(p) is convex because maximum of convex functions is convex

exact solution obtained with effort ≈ modest factor × least-squares effort

additional constraints: does adding (a) or (b) below complicate the problem?

(a) no more than half of total power is in any 10 lamps(b) no more than half of the lamps are on (pj > 0)

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• answer: with (a), still easy to solve; with (b), extremely difficult• moral: (untrained) intuition doesn’t always work; without the proper

background very easy problems can appear quite similar to very difficultproblems

6. Can you think of yet another way to solve the Illumination Problem exactly?

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References

• Primary Textbook: Convex Optimization by Stephen Boyd and

Lieven Vandenberghe, Cambridge University Press, 2004

http://www.stanford.edu/∼boyd/cvxbook

• Lectures on Modern Convex Optimization: Analysis, Algorithms,

and Engineering, by A. Ben-Tal and A. Nemirovski, SIAM, 2001

• Parallel and Distributed Computation: Numerical Methods by D.

Bertsekas and J. N. Tsitsiklis, 1st Edition, Prentice Hall, 1989

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