Univariate representations of solutions to generic polynomial complementarity problems
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.
Cite
@article{arxiv.2410.21810,
title = {Univariate representations of solutions to generic polynomial complementarity problems},
author = {Vu Trung Hieu and Alfredo Noel Iusem and Paul Hugo Schmölling and Akiko Takeda},
journal= {arXiv preprint arXiv:2410.21810},
year = {2025}
}