Conical Partition Algorithm for Maximizing the Sum of dc Ratios

Abstract

The following problem is considered in this paper: $$max_{x\in d\{\Sigma^m_{j=1}g_j(x)|h_j(x)\},}\, where\,g_j(x)\geq 0\, and\,h_j(x) > 0, j = 1,\ldots,m,$$ are d.c. (difference of convex) functions over a convex compact set D in R^n. Specifically, it is reformulated into the problem of maximizing a linear objective function over a feasible region defined by multiple reverse convex functions. Several favorable properties are developed and a branch-and-bound algorithm based on the conical partition and the outer approximation scheme is presented. Preliminary results of numerical experiments are reported on the efficiency of the proposed algorithm.