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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.
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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
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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
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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)
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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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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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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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