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. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 2 / 16
Cones Convex cones The set K R n is a convex cone if it is a convex set and for all x K and λ > 0 one has λx K. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 3 / 16
Convex cones Conic optimization problem Data: A convex cone K R n ; A linear operator A : R n R m ; Vectors c R n and b R m, and an inner product, on R n. Conic optimization problem inf { c, x : Ax = b}. x K Etienne de Klerk (Tilburg University) Conic optimization: examples and software 4 / 16
Choices for K Convex cones We consider the conic optimization problem for three choices of the cone K (or Cartesian products of cones of this type): Linear Programming (LP): K is the nonnegative orthant in R n : R n + := {x R n : x i 0 (i = 1,..., n)}, Second order cone programming (SOCP): K is the second order (Lorentz) cone: {[ ] } x : x R n, t R, t x. t Semidefinite programming (SDP): K is the cone of symmetric positive semidefinite matrices. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 5 / 16
Robust LP Examples for the second order cone We consider an LP problem with uncertain data. Robust LP Problem min c T x subject to ai T x b i (i = 1,..., m) a i E i (i = 1,..., m), where the E i are given ellipsoids: E i = {ā i + P i u : u 1}, with P i symmetric positive semidefinite. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 6 / 16
Examples for the second order cone Robust LP: SOCP formulation We had Notice that E i := {ā i + P i u : u 1}. a T i x b i a i E i ā T i x + P i x b i Robust LP Problem: SOCP reformulation subject to min c T x ā T i x + P i x b i (i = 1,..., m). Note that this is indeed an SOCP problem. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 7 / 16
References and info Examples for the second order cone The solutions of at least 13 of the 90 Netlib LP problems are meaningless if there is 0.01% uncertainty in the data entries! Solving the robust LP instead overcomes this difficulty. Ben-Tal, A., Nemirovski, A. Robust solutions of Linear Programming problems contaminated with 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, filter design... Etienne de Klerk (Tilburg University) Conic optimization: examples and software 8 / 16
Examples for the positive semidefinite cone Example: sum of squares polynomials Example Is p(x) := 2x 4 1 + 2x 3 1 x 2 x 2 1 x 2 2 + 5x 4 2 a sum of squared polynomials? YES, because p(x) = x 2 1 x 2 2 x 1 x 2 T 2 3 1 3 5 0 1 0 5 x 2 1 x 2 2 x 1 x 2. The 3 3 matrix (say M) is positive semidefinite and: and consequently p(x) = 1 2 M = L T L, L = 1 2 [ 2 3 1 0 1 3 ], ( 2x 2 1 3x2 2 ) 2 1 ( ) + x 1 x 2 + x 2 2 2 2 + 3x 1 x 2. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 9 / 16
Examples for the positive semidefinite cone Discussion: sum of squares polynomials The example illustrates the fact that deciding if a polynomial is a sum of squares 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 dynamical systems. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 10 / 16
Examples for the positive semidefinite cone Example: Lyapunov stability Definition The origin is asymptotically stable for a dynamical system ẋ(t) = f (x(t)), x(0) = x 0 if lim t x(t) = 0 whenever x 0 is sufficiently close to 0. A sufficient condition for stability is a nonnegative Lyapunov function V : R n R such that V (0) = 0 and V (x) T f (x) < 0 if x 0. Example (Parrilo): ẋ 1 (t) = x 2 (t) + 3 2 x 2 1 (t) 1 2 x 3 1 (t) ẋ 2 (t) = 3x 1 (t) x 2 (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. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 11 / 16
Examples for the positive semidefinite cone Example: Lyapunov stability (ctd.) contours of V (x); trajectories. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 12 / 16
Examples for the positive semidefinite cone Example: Nonnegative data fitting Etienne de Klerk (Tilburg University) Conic optimization: examples and software 13 / 16
References and info Examples for the positive semidefinite cone Lyapunov stability example from: P. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness 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 by nonnegative (trigonometric) polynomials and rational functions. Structural and Multidisciplinary Optimization, 35(4), 327-339. Other SDP applications include free material optimization, sensor network localization, low rank matrix completion,... Etienne de Klerk (Tilburg University) Conic optimization: examples and software 14 / 16
Examples for the positive semidefinite cone Free material optimization: wing design of the Airbus A380 Further reading M. Kočvara, M. Stingl and J. Zowe. Free material optimization: recent progress. Optimization, 57(1), 79 100, 2008. Etienne de Klerk (Tilburg University) Conic optimization: examples and software 15 / 16
Examples for the positive semidefinite cone 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 the LP/SOCP case): LP SOCP SDP n 10 6 10 8 10 5 10 6 10 3 10 3 m 10 6 10 5 10 3 Etienne de Klerk (Tilburg University) Conic optimization: examples and software 16 / 16