Convex generalized Nash equilibrium problems and polynomial optimization

Nie, Jiawang; Tang, Xindong

doi:10.1007/s10107-021-01739-7

Cited by 10 publications

(4 citation statements)

References 58 publications

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“…It took around 177 seconds to solve the complex KKT system. We would like to remark that in [15] and [39], only the first GNE was found, and the second to the fourth GNEs are new solutions found by our algorithm.…”

Section: Numerical Examplesmentioning

confidence: 93%

“…Comparison with existing methods. In this subsection, we compare the performance of Algorithm 3.1 with some existing methods for solving GNEPs, such as the augmented Lagrangian method (ALM) in [24], Gauss-Sedel method (GSM) in [40], the interior point method (IPM) in [12], and the semidefinite relaxation method (KKT-SDP) in [39]. We tested these methods on all GNEPs of polynomials in Examples 5.1-5.6.…”

Section: 1mentioning

confidence: 99%

“…GNEPs originated from economics in [2,11], and have been widely used in many other areas, such as telecommunications [1], supply chain [42] and machine learning [33]. There are plenty of interesting models formulated as GNEP of polynomials, and we refer to [14,15,38,39,42] for them.…”

Section: Introductionmentioning

confidence: 99%

“…There are some numerical algorithms to find GNEs, especially for convex GNEPs, such as the penalty method [3,15], the augmented Lagrangian method [24], the variational and quasi-variational inequality approach [13,16,19], the Nikaido-Isoda function approach [48,49], and the interior-point method on solving the KKT system [12]. Moreover, for convex GNEP of polynomials, a semidefinite relaxation method is introduced in [39], and it is extended to nonconvex rational GNEPs in [41]. The Gauss-Seidel method is studied in [40] for nonconvex GNEPs of polynomials.…”

Section: Introductionmentioning

confidence: 99%

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On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials

Lee¹,

Tang²

2022

Preprint

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The generalized Nash equilibrium problem (GNEP) is a kind of games to find strategies for a group of players such that each player's objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method.

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Section: Numerical Examplesmentioning

confidence: 93%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials

Lee¹,

Tang²

2022

Preprint

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Semidefinite games

Ickstadt,

Theobald,

Tsigaridas

2024

Int J Game Theory

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We introduce and study the class of semidefinite games, which generalizes bimatrix games and finite N-person games, by replacing the simplex of the mixed strategies for each player by a slice of the positive semidefinite cone in the space of real symmetric matrices. For semidefinite two-player zero-sum games, we show that the optimal strategies can be computed by semidefinite programming. Furthermore, we show that two-player semidefinite zero-sum games are almost equivalent to semidefinite programming, generalizing Dantzig’s result on the almost equivalence of bimatrix games and linear programming. For general two-player semidefinite games, we prove a spectrahedral characterization of the Nash equilibria. Moreover, we give constructions of semidefinite games with many Nash equilibria. In particular, we give a construction of semidefinite games whose number of connected components of Nash equilibria exceeds the long standing best known construction for many Nash equilibria in bimatrix games, which was presented by von Stengel in 1999.

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Polynomial Optimization Over Unions of Sets

Nie,

Zhang

2024

Vietnam J. Math.

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This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The application for computing (p, q)-norms of matrices is also presented.

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Convex generalized Nash equilibrium problems and polynomial optimization

Cited by 10 publications

References 58 publications

On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials

On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials

Semidefinite games

Polynomial Optimization Over Unions of Sets

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