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arXiv · 2410.21810

Univariate representations of solutions to generic polynomial complementarity problems

Vu Trung Hieu·Alfredo Noel Iusem·Paul Hugo Schmölling·Akiko Takeda
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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.

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Vu Trung Hieu, Alfredo Noel Iusem, Paul Hugo Schmölling, Akiko Takeda. 2024-10-29. Univariate representations of solutions to generic polynomial complementarity problems. https://doi.org/10.1007/s10957-025-02788-0

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