Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

• Published in: SIAM journal on optimization 2000, Vol.10 (3), p.750
• Main Authors: Kojima, Masakazu, Tunc¸el, Levent
• Format: Article
• Language: English
• Subjects: Applied mathematics; Euclidean space; Inequality; Integer programming; Linear programming; Optimization
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/S1052623498336450

Let F be a compact subset of the n-dimensional Euclidean space Rn represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets Ck (k = 1, 2, . . .) of Rn such that (a) the convex hull of $F \subseteq C_{k+1} \subseteq C_k$ (monotonicity), (b) $\cap_{k=1}^{\infty} C_k = \text{the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovasz--Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semi-infinite convex QOP relaxation proposed originally by Fujie and Kojima. Using this equivalence, we investigate some fundamental features of the two methods including (a) and (b) above.