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 Tunçel for approximating a convex relaxation of a compact subset F={x∈C0:p(x)≤0 (∀p(·)∈𝒫F)} of the n-dimensional Euclidean space Rⁿ. Here, C0 denotes a nonempty compact convex subset of Rⁿ, and 𝒫F a set of finitely or infinitely 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 ε, in terms of ε, the diameter of C0, the diameter of F, and some other quantities characterizing the Lipschitz continuity, the nonlinearity, and the nonconvexity of the set 𝒫F of quadratic functions.