Solving Reverse Convex Programming Problems Using a Generalized Cutting-Plane Method

Abstract

It is well known that each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is supremally generated by affine functions. More precisely, each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is the upper envelope of its affine minorants. In this paper, we propose an algorithm for solving reverse convex programming problems by using such a representation together with a generalized cutting-plane method. Indeed, by applying this representation, we solve a sequence of problems with a smaller feasible set, in which the reverse convex constraint is replaced by a still reverse convex but polyhedral constraint. Moreover, we prove that the proposed algorithm converges, under suitable assumptions, to an optimal solution of the original problem. This algorithm is coded in MATLAB language and is evaluated by some numerical examples.