Norm Bounds and Underestimators for Unconstrained Polynomial Integer Minimization

TL;DR

This paper develops new bounds and underestimators for unconstrained polynomial integer minimization, improving existing methods and demonstrating effectiveness in computational experiments, especially within branch and bound algorithms.

Contribution

It introduces a novel radius bound for integer minimizers and a new class of underestimators using sos programming, enhancing optimization techniques.

Findings

Radius bounds are tighter than previous literature.

New underestimators provide stronger lower bounds.

Experimental results show improved performance in branch and bound.

Abstract

We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer minimizers as well. In case this condition holds, we use sos programming to compute the radius of a p-norm ball which contains all integer minimizers. We prove that this radius is smaller than the radius known from the literature. Furthermore, we derive a new class of underestimators of the polynomial function. Using a Stellensatz from real algebraic geometry and again sos programming, we optimize over this class to get a strong lower bound on the integer minimum. Our radius and lower bounds are evaluated experimentally. They show a good performance, in particular within a branch and bound framework.