Univariate representations of solutions to generic polynomial complementarity problems

TL;DR

This paper introduces a method using squared slack variables to represent solutions of polynomial complementarity problems as roots of univariate polynomials, enabling efficient solution enumeration and approximation.

Contribution

It presents a novel approach to derive univariate representations of solutions for polynomial complementarity problems, including special solutions and approximate methods for infinite solution sets.

Findings

Univariate representations facilitate solution enumeration.

Symbolic computation effectively computes solutions for small problems.

Proposed methods handle problems with infinitely many solutions.

Abstract

By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper points out that, generically, this polynomial system has finitely many complex zeroes. In such a case, we use symbolic computation techniques to compute a univariate representation of the solution set. Consequently, univariate representations of special solutions, such as least-norm and sparse solutions, are obtained. After that, enumerating solutions boils down to solving problems governed by univariate polynomials. We also provide some experiments on small-scale problems with worst-case scenarios. At the end of the paper, we propose a method for computing approximate solutions to copositive polynomial complementarity problems that may have infinitely many solutions.