Bounds of the Solution Set to the Polynomial Complementarity Problem

• Published in: Journal of optimization theory and applications Vol. 203; no. 1; pp. 146 - 164
• Main Authors: Xu, Yang, Ni, Guyan, Zhang, Mengshi
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Lower bounds; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Optimization algorithms; Polynomials; Symmetry; Tensors; Theory of Computation; Upper bounds; 90C26; Polynomial complementarity problem; Tensor; Bounds of the solution set; 90C33
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-024-02484-5

In this paper, we investigate bounds of solution set of the polynomial complementarity problem. When a polynomial complementarity problem has a solution, we propose a lower bound of solution norm by entries of coefficient tensors of the polynomial. We prove that the proposing lower bound is larger than some existing lower bounds appeared in tensor complementarity problems and polynomial complementarity problems. When the solution set of a polynomial complementarity problem is nonempty, and the coefficient tensor of the leading term of the polynomial is an
R
0
-tensor, we propose a new upper bound of solution norm of the polynomial complementarity problem by a quantity defining by an optimization problem. Furthermore, we prove that when coefficient tensors of the polynomial are partially symmetric, the proposing lower bound formula with respect to tensor tuples reaches the maximum value, and the proposing upper bound formula with respect to tensor tuples reaches the minimum value. Finally, by using such partial symmetry, we obtain bounds of solution norm by coefficients of the polynomial.