Finding the Least Element of a Nonnegative Solution Set of a Class of Polynomial Inequalities
Abstract.
In this paper, we consider the least element problem of a nonnegative solution set of a system of inequalities defined by a homogeneous polynomial mapping and a vector. In the set under consideration, the homogeneous polynomial mapping is defined by a tensor. When the tensor involved is square, the set under consideration is just the feasible region of the tensor complementarity problem (TCP). We first introduce the concept of the generalized \(Z\) -tensor, which is a natural generalization of the \(Z\) -tensor. Then, under the assumption that the considered set is nonempty and the tensor involved is a generalized \(Z\) -tensor, we propose an iterative method for finding the least element of the considered set. Specifically, by recognizing the position indices of positive components in the least element continuously, we solve a series of lower-dimensional system of tensor equations corresponding to these indices and prove that the least element of the set can be obtained within finite step iterations. When the tensor involved is square, the least element obtained is also a solution of the TCP. Compared with the existing methods for finding the least-element solution of the TCP with a \(Z\) -tensor, our method does not require any additional assumptions and has lower computational cost. Preliminary numerical experiments show that the proposed method is effective.
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Information & Authors
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Published In
SIAM Journal on Matrix Analysis and Applications
Pages: 530 - 558
DOI: 10.1137/22M1476733
ISSN (online): 1095-7162
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 8 February 2022
Accepted: 21 November 2022
Published online: 11 May 2023
Keywords
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Authors
Funding Information
Science and Technology Program of Guangzhou: 202002030280
National Natural Science Foundation of China (NSF): 11871051, 11901600, 12171357
Funding: This work was funded by the National Natural Science Foundation of China under grants 11871051, 11901600, and 12171357 and by the Science and Technology Program of Guangzhou under grant 202002030280.
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