rob-sdp - SIAM J OPTIM Vol 9 No 1 pp 3352 c 1998 Society...

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ROBUST SOLUTIONS TO UNCERTAIN SEMIDEFINITE PROGRAMS * LAURENT EL GHAOUI , FRANCOIS OUSTRY , AND HERV ´ E LEBRET SIAM J. O PTIM . c ± 1998 Society for Industrial and Applied Mathematics Vol. 9, No. 1, pp. 33–52 Abstract. In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek “robust” solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is “full,” our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (H¨older-stable) with respect to the unperturbed problem’s data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming. Key words. convex optimization, semidefinite programming, uncertainty, robustness, regular- ization AMS subject classifications. 93B35, 49M45, 90C31, 93B60 PII. S1052623496305717 Notation. For a matrix X , k X k denotes the largest singular value. If X is square, X ± 0 (resp., X ² 0) means X is symmetric and positive semidefinite (resp., definite). For X ± 0, X 1 / 2 denotes the symmetric square root of X . The notation I p denotes the p × p identity matrix; the subscript is omitted when it can be inferred from context. 1. Introduction. A semidefinite program (SDP) consists of minimizing a linear objective under a linear matrix inequality (LMI) constraint; precisely, P 0 : minimize c T x subject to F ( x )= F 0 + m X i =1 x i F i ± 0 , (1) where c R m -{ 0 } and the symmetric matrices F i = F T i R n × n ,i =0 ,...,m , are given. SDPs are convex optimization problems and can be solved in polynomial time with, e.g., primal-dual interior-point methods [24, 35, 26, 19, 2]. SDPs include linear programs and convex quadratically constrained quadratic programs, and arise in a wide range of engineering applications; see, e.g., [12, 1, 35, 22]. In the SDP (1), the “data” consist of the objective vector c and the matrices F 0 ,...,F m . In practice, these data are subject to uncertainty. An extensive body of work has concentrated on the sensitivity issue, in which the perturbations are assumed to be infinitesimal, and regularity of optimal values and solution(s), as functions of the data matrices, is studied. Recent references on sensitivity analysis include [30, 31, 10] for general nonlinear programs, [33] for semi-infinite programs, and [32] for semidefinite programs.
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This note was uploaded on 09/05/2009 for the course EECS 227A taught by Professor Martinwainwright during the Fall '04 term at Berkeley.

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rob-sdp - SIAM J OPTIM Vol 9 No 1 pp 3352 c 1998 Society...

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