Complexity Analysis of Successive Convex Relaxation Methods for Nonconvex Sets

This paper discusses computational complexity of conceptual successive convex relaxation methods proposed by Kojima and Tunn cel for approximating a convex relaxation of a compact subset F = fx 2 C 0 : p(x) 0 (8p() 2 P F)g of the n-dimensional Euclidean space R n. Here C 0 denotes a nonempty compact convex subset of R n , and P F a set of nitely or innnitely many quadratic functions. We evaluate the number of iterations which the successive convex relaxation methods require to attain a convex relaxation of F with a given accuracy , in terms of , the diameter of C 0 , the diameter of F, and some other quantities characterizing the Lipschitz continuity, the nonlinearity and the nonconvexity of the set P F of quadratic functions.