
Journal of Convex Analysis 25 (2018), No. 4, 11591182 Copyright Heldermann Verlag 2018 Tight SDP Relaxations for a Class of Robust SOSConvex Polynomial Programs without the Slater Condition Thai Doan Chuong School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia chuongthaidoan@yahoo.com Vaithilingam Jeyakumar School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia v.jeyakumar@unsw.edu.au We examine a robust SOSconvex polynomial program under a general class of bounded uncertainty, called intersection ellipsoidal uncertainty, which is described by the intersection of a family of ellipsoids and covers many commonly used uncertainty classes of robust optimization. The class of SOSconvex polynomials is a numerically tractable subclass of convex polynomials and, in particular, contains convex quadratic functions and convex separable polynomials. We present a conical linear program with a coupled sumofsquares polynomial and linear matrix inequality constraints as its relaxation problem and show that the relaxation problem can equivalently be reformulated as a semidefinite linear program (SDP). Under a mild wellposedness condition, we establish that the socalled SDP relaxation is tight in the sense that the optimal values of the robust SOSconvex polynomial program and its relaxation problem are equal. We also show, under a general regularity condition, that the SDP relaxation is exact in the sense that the relaxation problem not only shares the same optimal value with the robust SOSconvex polynomial program but also attains its optimum, extending the corresponding known results in the robust optimization literature to the general class of intersection ellipsoidal uncertainty. Keywords: Robust optimization, semiinfinite convex program, convex polynomial, semidefinite linear program, SDP relaxation. MSC: 49K99, 65K10, 90C29, 90C46 [ Fulltextpdf (163 KB)] for subscribers only. 