New Partitioning Method for a Class of Nonconvex Optimization Problems

We consider the problem min{f(x, y): g_i(x, y) ≤ 0, i = 1, …, m, x ∈ X, y ∈ Y} where f and the g_i are lower semicontinuous and convex in y for fixed x but not convex jointly in (x, y), X is a compact subset of Rⁿ, Y is a closed convex set in R^m. In order to decompose this problem into subproblems, each depending either on the x-variables alone or the y-variables alone, we propose a new partitioning method which does not require the Benders-Geoffrion's condition on the structure of joint constraints and the objective function. The relaxed master problems generated by our partitioning method are d.c programs and they can be systematically solved by recent algorithms.