Some bounds of solutions to the polynomial complementarity problem

Jun He; Shi-Liang Wu; Dongsheng Luo; Qingyu Zeng; Jun He; Shi-Liang Wu; Dongsheng Luo; Qingyu Zeng

doi:10.3934/jimo.2026098

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 6: 2683-2696. doi: 10.3934/jimo.2026098

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Research article

Some bounds of solutions to the polynomial complementarity problem

1.
School of Mathematics, Zunyi Normal College, Zunyi, Guizhou, 563006, P.R. China
2.
Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming, Yunnan, 650500, P.R. China

Received: 16 January 2026 Revised: 28 April 2026 Accepted: 07 May 2026 Published: 13 May 2026
90C33, 90C26

The polynomial complementarity problem (PCP) is to find a vector $ {\bf x}\in \mathbb{R}^n_+ $ such that $ \sum_{h = 1}^{m-1}\mathcal{A}_h {\bf x}^{m-h} +{\bf q} \ge{\bf 0}, {\bf x}^{\top}\big(\sum_{h = 1}^{m-1}\mathcal{A}_h {\bf x}^{m-h}+{\bf q}\big) = 0 $. In this paper, we further investigate the lower bound for the solution set of the PCP with $ {\bf q} \in \mathbb{R}^n_{++} $. When $ \mathcal{A}_1 $ is an $ R_0 $-tensor, we present an improved upper bound for the solution set of the PCP, which is tighter than the bound given by Xu et al. in [Bounds of the solution set to the polynomial complementarity problem. J. Optim. Theory Appl. 203 (2024) 146-164]. Finally, we prove that the proposed lower and upper bounds for the solution set of the PCP with partially symmetric tensor tuples are attainable. Numerical examples are given to show the efficiency of the proposed results.
- upper bounds,
- lower bounds,
- tensor,
- polynomial complementarity problem
Citation: Jun He, Shi-Liang Wu, Dongsheng Luo, Qingyu Zeng. Some bounds of solutions to the polynomial complementarity problem[J]. Journal of Industrial and Management Optimization, 2026, 22(6): 2683-2696. doi: 10.3934/jimo.2026098

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Abstract

The polynomial complementarity problem (PCP) is to find a vector $ {\bf x}\in \mathbb{R}^n_+ $ such that $ \sum_{h = 1}^{m-1}\mathcal{A}_h {\bf x}^{m-h} +{\bf q} \ge{\bf 0}, {\bf x}^{\top}\big(\sum_{h = 1}^{m-1}\mathcal{A}_h {\bf x}^{m-h}+{\bf q}\big) = 0 $. In this paper, we further investigate the lower bound for the solution set of the PCP with $ {\bf q} \in \mathbb{R}^n_{++} $. When $ \mathcal{A}_1 $ is an $ R_0 $-tensor, we present an improved upper bound for the solution set of the PCP, which is tighter than the bound given by Xu et al. in [Bounds of the solution set to the polynomial complementarity problem. J. Optim. Theory Appl. 203 (2024) 146-164]. Finally, we prove that the proposed lower and upper bounds for the solution set of the PCP with partially symmetric tensor tuples are attainable. Numerical examples are given to show the efficiency of the proposed results.

References

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