On Global Maximization of Bounded Hessian Functions over Strongly Convex Domains

TL;DR

This paper introduces two frameworks that reduce the problem of globally maximizing bounded Hessian functions over strongly convex domains to convex optimization, with applications including polynomial-time algorithms for specific geometric and combinatorial problems.

Contribution

It presents improved algorithms and frameworks for global maximization of bounded Hessian functions, including polynomial-time solutions for certain geometric and subset sum problems.

Findings

Explicit polynomial-time algorithm for farthest point in finite intersection of n-disks when outside convex hull

Second framework characterizes the furthest point when inside the convex hull, despite being computationally hard in general

Application to solve some subset sum problems with real entries in polynomial time

Abstract

In this paper we present two frameworks in which global maximization of a bounded hessian function over a strongly convex set can be reduced to convex optimization. The first presented framework is a continuation of one of our previous papers [11]. We improve the results and give an explicit algorithm for the computation in polynomial time of the farthest point in a finite intersection of n-disks to $C \in \mathbb{R}^{n \times 1}$ under the requirement that $C$ does not belong to the convex hull of the centers of the n-disks. Finally, in order to overcome this limitation we present a second framework which characterizes the furthest in the finite intersection of n-disks with $C$ in the convex hull. Unfortunately this second framework requires the ability to decide if a polytope that we define is included in the intersection, which is hard in general. However, as a particular application of our second framework we are able solve in P time some instances of the subset sum problem with real entries: given a set of real numbers decide if there is a subset which adds up to zero.