Conic optimization: examples and software - Tilburg University · Conic optimization: examples and...

Conic optimization: examples and software

Etienne de Klerk

Tilburg University, The Netherlands

Etienne de Klerk (Tilburg University) Conic optimization: examples and software 1 / 16

Outline

Conic optimization

Second order cone optimization example: robust linear programming;

Semidefinite programming examples: Lyapunov stability and data fitting;

Software.


Convex cones

Cones

The set K ⊂ Rn is a convex cone if it is a convex set and for all x ∈ K and λ > 0one has λx ∈ K .


Convex cones

Conic optimization problem

Data:

A convex cone K ⊂ Rn;

A linear operator A : Rn → Rm;

Vectors c ∈ Rn and b ∈ Rm, and an inner product 〈·, ·〉 on Rn.

Conic optimization problem

infx∈K{〈c , x〉 : Ax = b} .


Convex cones

Choices for K

We consider the conic optimization problem for three choices of the cone K (orCartesian products of cones of this type):

Linear Programming (LP): K is the nonnegative orthant in Rn:

Rn+ := {x ∈ Rn : xi ≥ 0 (i = 1, . . . , n)} ,

Second order cone programming (SOCP): K is the second order (Lorentz)cone: {[

xt

]: x ∈ Rn, t ∈ R, t ≥ ‖x‖

}.

Semidefinite programming (SDP): K is the cone of symmetric positivesemidefinite matrices.


Examples for the second order cone

Robust LP

We consider an LP problem with ‘uncertain’ data.

Robust LP Problem

min cT x

subject toaTi x ≤ bi (i = 1, . . . ,m) ∀ai ∈ Ei (i = 1, . . . ,m),

where the Ei are given ellipsoids:

Ei = {ai + Piu : ‖u‖ ≤ 1},

with Pi symmetric positive semidefinite.



Robust LP: SOCP formulation

We hadEi := {ai + Piu : ‖u‖ ≤ 1}.

Notice thataTi x ≤ bi ∀ai ∈ Ei ⇐⇒ aTi x + ‖Pix‖ ≤ bi

Robust LP Problem: SOCP reformulation

min cT x

subject toaTi x + ‖Pix‖ ≤ bi (i = 1, . . . ,m).

Note that this is indeed an SOCP problem.



References and info

The solutions of at least 13 of the 90 Netlib LP problems are meaningless ifthere is 0.01% uncertainty in the data entries! Solving the robust LP insteadovercomes this difficulty.

Ben-Tal, A., Nemirovski, A. Robust solutions of Linear Programming problems contaminatedwith uncertain data. Math. Progr. 88 (2000), 411–424.

More SOCP examples in the online paper:

M. Lobo, L. Vandenberghe, S. Boyd, H. Lebret, Applications of second-order cone programming.Linear Algebra and its Applications, 1998.

Applications include robust least squares problems, portfolio selection, filterdesign ...


Examples for the positive semidefinite cone

Example: sum of squares polynomials

Example

Is p(x) := 2x41 + 2x3

1 x2 − x21 x

22 + 5x4

2 a sum of squared polynomials?

YES, because

p(x) =

x21

x22

x1x2

T 2 −3 1−3 5 0

1 0 5

x21

x22

x1x2

.The 3× 3 matrix (say M) is positive semidefinite and:

M = LTL, L =1√2

[2 −3 10 1 3

],

and consequently

p(x) =1

2

(2x2

1 − 3x22 + x1x2

)2+

1

2

(x2

2 + 3x1x2

)2.



Discussion: sum of squares polynomials

The example illustrates the fact that deciding if a polynomial is a sum ofsquares is equivalent to an SDP problem;

This has application in polynomial optimization problems, ...

... data fitting using nonnegative or monotone polynomials,

... and finding polynomial Lyapunov functions to prove stability of dynamicalsystems.



Example: Lyapunov stability

Definition

The origin is asymptotically stable for a dynamical system

x(t) = f (x(t)), x(0) = x0

if limt→∞ x(t) = 0 whenever x0 is sufficiently close to 0. A sufficient condition forstability is a nonnegative Lyapunov function V : Rn → R such that V (0) = 0 and∇V (x)T f (x) < 0 if x 6= 0.

Example (Parrilo):

x1(t) = −x2(t) +3

2x2

1 (t)− 1

2x3

1 (t)

x2(t) = 3x1(t)− x2(t).

Using SDP, one may find a degree 4 polynomial V to prove stability, ...

... where both V (x) and −∇V (x)T f (x) are sums of squares.



Example: Lyapunov stability (ctd.)

−− contours of V (x);

— trajectories.



Example: Nonnegative data fitting



References and info

Lyapunov stability example from:

P. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods inRobustness and Optimization, PhD thesis, Caltech, May 2000.

Data fitting example from:

Siem, A.Y.D., Klerk, E. de, and Hertog, D. den (2008). Discrete least-norm approximation bynonnegative (trigonometric) polynomials and rational functions. Structural and MultidisciplinaryOptimization, 35(4), 327-339.

Other SDP applications include free material optimization, sensor networklocalization, low rank matrix completion, ...



Free material optimization: wing design of the Airbus A380

Further reading

M. Kocvara, M. Stingl and J. Zowe. Free material optimization: recent progress. Optimization,57(1), 79 – 100, 2008.



Software

Software that implements interior point methods for conic programming:

Commercial LP solvers: CPLEX, MOSEK, XPRESS-MP, ...

SOCP solvers: MOSEK, LOQO, SeDuMi

SDP solvers: SDPT3, SeDuMi, CSDP, SDPA ...

Sizes of problems that can be solved in ”reasonable time” (sparse data in theLP/SOCP case):

LP SOCP SDPn 106–108 105–106 103 × 103

m 106 105 103


Conic optimization: examples and software - Tilburg University · Conic optimization: examples and...

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