A modification piecewise convexification method with a classification strategy for box-constrained non-convex optimization programs

Abstract

This paper presents a piecewise convexification method with a box classification strategy to approximate the entire globally optimal solution set of non-convex optimization problems with box constraints. First, the box classification strategy is proposed based on the convexity of the objective function on the sub-boxes, which helps to reduce the number of box divisions and improve the computational efficiency. At the same time, we construct the piecewise convexification problem of the original nonconvex optimization problem by applying the α-based Branch-and-Bound (αBB) method, and we define the (approximate) solution set of the piecewise convexification problem based on the result of classifying the sub-boxes. Then, it is deduced that the globally optimal solution set can be approximated by the (approximate) solution set of the piecewise convexification problem. Finally, a piecewise convexification algorithm is proposed that includes a new subset selection technique for division and two new termination tests. The results of our experiments demonstrate the effectiveness and general superiority of our approach over the competition.