Generation of interior points and polyhedral representations of cones in ${\mathbb {R}}^N$ cut by ${\varvec{M}}$ planes sharing a common point

Abstract

The following paper describes a method for solving certain non-convex minimization problems. These problems contain properties where the feasible region may be represented as a union of convex cones. This set of cones is generated by the intersection of finitely many half-spaces passing through a defined origin. Globally the problem is non-convex, but it is locally convex within the interior of the generated cones whereas the objective function tends to infinity elsewhere. The following algorithm results in a polyhedral representation of all of the cones and an interior point in each of them. In order to solve the problem globally, a convex optimization algorithm is required to be initialized at each interior point and to converge to the local minimum over the corresponding cone. The best of these solutions is the global optimum. A study of the time complexity is presented herewith along with a detailed computational analysis.