A Polynomial Optimization Framework for Polynomial Quasi-variational Inequalities with Moment-sos Relaxations

Xindong Tang; Min Zhang; Wenzhi Zhong

doi:10.3934/naco.2024054

A Polynomial Optimization Framework for Polynomial Quasi-variational Inequalities with Moment-sos Relaxations

Xindong Tang^*
, Min Zhang
, Wenzhi Zhong

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

Abstract

We consider quasi-variational inequality problems (QVI) given by polynomial functions. By applying Lagrange multiplier expressions, we formulate polynomial optimization problems whose minimizers are KKT points for the QVI. Then, feasible extensions are exploited to preclude KKT points that are not solutions. Moment-SOS relaxations are incorporated to solve the polynomial optimization problems in our methods. Under certain conditions, our approach guarantees to find a solution to the QVI or detect the nonexistence of solutions.

Original language	English
Pages (from-to)	61-83
Number of pages	23
Journal	Numerical Algebra, Control and Optimization
Volume	16
Early online date	Dec 2024
DOIs	https://doi.org/10.3934/naco.2024054
Publication status	Published - Mar 2026

User-Defined Keywords

Lagrange multiplier expression
Moment-SOS hierarchy
Quasi-variational inequality
Polynomial optimization

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10.3934/naco.2024054

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title = "A Polynomial Optimization Framework for Polynomial Quasi-variational Inequalities with Moment-sos Relaxations",

abstract = "We consider quasi-variational inequality problems (QVI) given by polynomial functions. By applying Lagrange multiplier expressions, we formulate polynomial optimization problems whose minimizers are KKT points for the QVI. Then, feasible extensions are exploited to preclude KKT points that are not solutions. Moment-SOS relaxations are incorporated to solve the polynomial optimization problems in our methods. Under certain conditions, our approach guarantees to find a solution to the QVI or detect the nonexistence of solutions.",

keywords = "Lagrange multiplier expression, Moment-SOS hierarchy, Quasi-variational inequality, Polynomial optimization",

author = "Xindong Tang and Min Zhang and Wenzhi Zhong",

note = "Xindong Tang is supported by the NSFC Young Scientists Fund, grant number [12301407]. Min Zhang is supported by the National Natural Science Foundation of China Youth Fund Project, grant number [12101598]. Publisher Copyright: {\textcopyright} 2026 American Institute of Mathematical Sciences. All rights reserved.",

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month = mar,

doi = "10.3934/naco.2024054",

language = "English",

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T1 - A Polynomial Optimization Framework for Polynomial Quasi-variational Inequalities with Moment-sos Relaxations

AU - Tang, Xindong

AU - Zhang, Min

AU - Zhong, Wenzhi

N1 - Xindong Tang is supported by the NSFC Young Scientists Fund, grant number [12301407]. Min Zhang is supported by the National Natural Science Foundation of China Youth Fund Project, grant number [12101598]. Publisher Copyright: © 2026 American Institute of Mathematical Sciences. All rights reserved.

PY - 2026/3

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N2 - We consider quasi-variational inequality problems (QVI) given by polynomial functions. By applying Lagrange multiplier expressions, we formulate polynomial optimization problems whose minimizers are KKT points for the QVI. Then, feasible extensions are exploited to preclude KKT points that are not solutions. Moment-SOS relaxations are incorporated to solve the polynomial optimization problems in our methods. Under certain conditions, our approach guarantees to find a solution to the QVI or detect the nonexistence of solutions.

AB - We consider quasi-variational inequality problems (QVI) given by polynomial functions. By applying Lagrange multiplier expressions, we formulate polynomial optimization problems whose minimizers are KKT points for the QVI. Then, feasible extensions are exploited to preclude KKT points that are not solutions. Moment-SOS relaxations are incorporated to solve the polynomial optimization problems in our methods. Under certain conditions, our approach guarantees to find a solution to the QVI or detect the nonexistence of solutions.

KW - Lagrange multiplier expression

KW - Moment-SOS hierarchy

KW - Quasi-variational inequality

KW - Polynomial optimization

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JF - Numerical Algebra, Control and Optimization

ER -

A Polynomial Optimization Framework for Polynomial Quasi-variational Inequalities with Moment-sos Relaxations

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