Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

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 ⊆ Ck+1 ⊆ Ck (monotonicity), (b) ∩k=1Ck = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovász–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 semiinfinite 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.