Global Minimization of a Linearly Constrained Concave Function by Partition of Feasible Domain

Abstract

The problem of minimizing a concave function subject to linear inequality constraints may have many local solutions. Therefore, finding the global constrained minimum is a computationally difficult problem. A computational method is described which finds the global minimum of a smooth concave function over a polyhedron in Rn. The feasible domain is partitioned into a rectangular domain, which can be excluded from further consideration, and r ≤ 2n subdomains, at least one of which contains the global minimum. A known algorithm can be applied sequentially (or in parallel) to each of these r subdomains to compute the global minimum. A method is also presented (Appendix B) for the construction of nontrivial test problems for which the global minimum point is known. Given an arbitrary polyhedron and a selected vertex, it is shown how to determine a concave quadratic function (generally with many local minima) with its global minimum at the selected vertex.