A variational inequality approach to Berge equilibrium problems

Mengkezhula Sagaarenchen; Battur Gompil; Batbileg Sukhee; Ankhbayar Gelegbadam; Enkhbat Rentsen; Mengkezhula Sagaarenchen; Battur Gompil; Batbileg Sukhee; Ankhbayar Gelegbadam; Enkhbat Rentsen

doi:10.3934/jimo.2026207

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 12: 6150-6165. doi: 10.3934/jimo.2026207

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Research article

A variational inequality approach to Berge equilibrium problems

1.
Department of Statistics and Applied Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia
2.
Institute of Mathematics and Digital Technology, Academy of Sciences of Mongolia, Ulaanbaatar, Mongolia

Received: 30 June 2026 Revised: 24 October 2026 Accepted: 27 October 2026 Published: 04 November 2026
91A10, 91A06, 90C33, 90C46

In this work, we study the Berge equilibrium problem (BEP) using a variational inequality (Ⅵ) framework. We establish an exact equivalence between the classical BEP and a corresponding Ⅵ problem under a pairwise symmetry condition. Furthermore, we extend this framework to generalized Berge equilibrium problems with shared (jointly convex) constraints, proving that the proposed Ⅵ formulation provides a rigorous sufficient condition for generalized Berge equilibria under appropriate assumptions. Illustrative examples are presented to demonstrate the theoretical findings.
- Berge equilibrium,
- variational inequalities,
- shared constraints,
- pairwise symmetry condition,
- generalized Berge equilibrium
Citation: Mengkezhula Sagaarenchen, Battur Gompil, Batbileg Sukhee, Ankhbayar Gelegbadam, Enkhbat Rentsen. A variational inequality approach to Berge equilibrium problems[J]. Journal of Industrial and Management Optimization, 2026, 22(12): 6150-6165. doi: 10.3934/jimo.2026207

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Abstract

In this work, we study the Berge equilibrium problem (BEP) using a variational inequality (Ⅵ) framework. We establish an exact equivalence between the classical BEP and a corresponding Ⅵ problem under a pairwise symmetry condition. Furthermore, we extend this framework to generalized Berge equilibrium problems with shared (jointly convex) constraints, proving that the proposed Ⅵ formulation provides a rigorous sufficient condition for generalized Berge equilibria under appropriate assumptions. Illustrative examples are presented to demonstrate the theoretical findings.

References

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