D.C. programming approach to generalized Nash equilibrium with quadratic objective functions

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

doi:10.3934/jimo.2026031

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 2: 860-879. doi: 10.3934/jimo.2026031

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Research article

D.C. programming approach to generalized Nash equilibrium with quadratic objective functions

1.
Department of Applied Mathematics, National University of Mongolia, Ikh surguuliin gudamj-1, Ulaanbaatar, 14201, Mongolia
2.
Institute of Mathematics and Digital Technology, Academy of Sciences of Mongolia, Peace avenue 54b, Ulaanbaatar, 13330, Mongolia

Received: 14 October 2025 Revised: 01 December 2025 Accepted: 22 December 2025 Published: 07 January 2026
90C20, 90A05, 49K99

In this paper, we present a new approach for solving the generalized Nash equilibrium problem (GNEP), where players have nonconvex quadratic objective functions and are subject to jointly convex shared constraints. We showed that this problem can be reformulated as a nonconvex unconstrained optimization problem based on a regularized Nikaido-Isoda function. This reformulation reduces to a difference of two convex functions (D.C.) programming problem, which enables us to apply local and global search methods developed in A. S. Strekalovsky. Numerical results are presented.
- nonconvex games,
- Nikaido-Isoda function,
- gap functions,
- D.C. optimization,
- local and global search methods
Citation: Battur Gompil, Mengkezhula Sagaarenchen, Batbileg Sukhee, Enkhbat Rentsen. D.C. programming approach to generalized Nash equilibrium with quadratic objective functions[J]. Journal of Industrial and Management Optimization, 2026, 22(2): 860-879. doi: 10.3934/jimo.2026031

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Abstract

In this paper, we present a new approach for solving the generalized Nash equilibrium problem (GNEP), where players have nonconvex quadratic objective functions and are subject to jointly convex shared constraints. We showed that this problem can be reformulated as a nonconvex unconstrained optimization problem based on a regularized Nikaido-Isoda function. This reformulation reduces to a difference of two convex functions (D.C.) programming problem, which enables us to apply local and global search methods developed in A. S. Strekalovsky. Numerical results are presented.

References

[1]	G. Debreu, A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA, 38 (1952), 886–893. https://doi.org/10.1073/pnas.38.10.886 doi: 10.1073/pnas.38.10.886
[2]	F. Facchinei, C. Kanzow, Generalized Nash equilibrium problems, Ann. Oper. Res., 175 (2010), 177–211. https://doi.org/10.1007/s10479-009-0653-x doi: 10.1007/s10479-009-0653-x
[3]	A. Fischer, M. Herrich, K. Schönefeld, Generalized Nash equilibrium problems–-recent advances and challenges, Pesq. Oper., 34 (2014), 521–558. https://doi.org/10.1590/0101-7438.2014.034.03.0521 doi: 10.1590/0101-7438.2014.034.03.0521
[4]	A. Dreves, C. Kanzow, Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems, Comput. Optim. Appl., 50 (2011), 23–48. https://doi.org/10.1007/s10589-009-9314-x doi: 10.1007/s10589-009-9314-x
[5]	N. Harms, Primal and Dual Gap Functions for Generalized Nash Equilibrium Problems and Quasi-Variational Inequalities, PhD dissertation, Würzburg University, 2014.
[6]	F. Facchinei, C. Kanzow, Penalty methods for the solution of generalized Nash equilibrium problems, SIAM J. Optim., 20 (2010), 2228–2253. https://doi.org/10.1137/090749499 doi: 10.1137/090749499
[7]	K. Nabetani, P. Tseng, M. Fukushima, Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints, Comput. Optim. Appl., 48 (2011), 423–452. https://doi.org/10.1007/s10589-009-9256-3 doi: 10.1007/s10589-009-9256-3
[8]	L. Altangerel, G. Battur, Perturbation approach to generalized Nash equilibrium problems with shared constraints, Optim. Lett., 6 (2012), 1379–1391. https://doi.org/10.1007/s11590-012-0510-8 doi: 10.1007/s11590-012-0510-8
[9]	L. Altangerel, G. Battur, An exact penalty approach and conjugate duality for generalized Nash equilibrium problems with coupling and shared constraints, Bull. Irkutsk State Univ., Ser. Math., 32 (2020), 3–16. https://doi.org/10.26516/1997-7670.2020.32.3 doi: 10.26516/1997-7670.2020.32.3
[10]	C. Malfatti, Memoria–-sopra un problema stereotomico, Mem. Mat. Fis. Soc. Ital. Sci., 10 (1803), 235–244.
[11]	R. Enkhbat, G. Battur, Generalized Nash equilibrium problem based on Malfatti's problem, Numer. Algebra Control Optim., 11 (2021), 209–220. https://doi.org/10.3934/naco.2020022 doi: 10.3934/naco.2020022
[12]	G. Battulga, L. Altangerel, G. Battur, Loan interest rate Nash models with solvency constraints in the banking sector, Optim. Methods Softw., 36 (2021), 891–908. https://doi.org/10.1080/10556788.2021.1891537 doi: 10.1080/10556788.2021.1891537
[13]	G. Battulga, L. Altangerel, G. Battur, An extension of one-period Nash equilibrium model in non-life insurance markets, Appl. Math., 9 (2018), 1339–1350. https://doi.org/10.4236/am.2018.912087 doi: 10.4236/am.2018.912087
[14]	K. Wang, W. S. Jia, A new intelligent algorithm for solving generalized Nash equilibrium problem, Alexandria Eng. J., 123 (2025), 17–28. https://doi.org/10.1016/j.aej.2025.03.044 doi: 10.1016/j.aej.2025.03.044
[15]	L. P. Liu, W. S. Jia, The value function with regret minimization algorithm for solving the Nash equilibrium of multi-agent stochastic game, Int. J. Comput. Intell. Syst., 14 (2021), 1633–1641. https://doi.org/10.2991/ijcis.d.210520.001 doi: 10.2991/ijcis.d.210520.001
[16]	Y. B. Li, W. S. Jia, L. P. Liu, Cooperative emergence induced by asymmetric punishment and strategy persistence mechanisms in multilayer networks, Commun. Nonlinear Sci. Numer. Simul., 152 (2026), 109187. https://doi.org/10.1016/j.cnsns.2025.109187 doi: 10.1016/j.cnsns.2025.109187
[17]	G. Battur, S. Batbileg, R. Enkhbat, A global optimization approach to Berge equilibrium based on a regularized function, Optim. Lett., (2024), 1–12. https://doi.org/10.1007/s11590-024-02141-w
[18]	S. Mengkezhula, S. Batbileg, R. Enkhbat, G. Battur, D.C. optimization approach for finding Berge equilibrium in bimatrix game, Numer. Algebra Control Optim., (2024), 1–16. https://doi.org/10.3934/naco.2024046
[19]	J. S. Pang, G. Scutari, Nonconvex games with side constraints, SIAM J. Optim., 21 (2011), 1491–1522. https://doi.org/10.1137/100811787 doi: 10.1137/100811787
[20]	I. Minarchenko, Search of Nash equilibrium in quadratic nonconvex game with weighted potential, in: S. Belim et al. (eds.), OPTA-SCL, Omsk, Russia, 2018. Available at: https://api.semanticscholar.org/CorpusID: 53608177.
[21]	A. S. Strekalovsky, Elements of Nonconvex Optimization, Nauka, Novosibirsk, 2003 (in Russian).
[22]	H. A. Le Thi, T. P. Dinh, D.C. programming and DCA: thirty years of developments, Math. Program., Ser. B, 169 (2018), 5–68. https://doi.org/10.1007/s10107-018-1235-y doi: 10.1007/s10107-018-1235-y
[23]	H. A. Le Thi, T. P. Dinh, N. D. Yen, Behavior of DCA sequences for solving the trust-region subproblem, J. Glob. Optim., 53 (2012), 317–329. https://doi.org/10.1007/s10898-011-9696-z doi: 10.1007/s10898-011-9696-z
[24]	H. A. Le Thi, T. P. Dinh, The D.C. (difference of convex functions) programming and DCA revisited with D.C. models of real world nonconvex optimization problems, Ann. Oper. Res., 133 (2005), 23–46. https://doi.org/10.1007/s10479-004-5022-1 doi: 10.1007/s10479-004-5022-1
[25]	H. A. Le Thi, V. N. Huynh, T. P. Dinh, Convergence analysis of difference-of-convex algorithm with subanalytic data, J. Optim. Theory Appl., 179 (2018), 103–126. https://doi.org/10.1007/s10957-018-1345-y doi: 10.1007/s10957-018-1345-y
[26]	N. T. Hoang, Linear convergence of a type of iterative sequences in nonconvex quadratic programming, J. Math. Anal. Appl., 423 (2014), 1311–1319. https://doi.org/10.1016/j.jmaa.2014.10.048 doi: 10.1016/j.jmaa.2014.10.048
[27]	H. Anna, K. Christian, Optimization reformulations of the generalized Nash equilibrium problem using Nikaido–Isoda-type functions, Comput. Optim. Appl., 43 (2009), 353–377. https://doi.org/10.1007/s10589-007-9145-6 doi: 10.1007/s10589-007-9145-6
[28]	N. T. Hoang, Convergence rate of the Pham Dinh–Le Thi algorithm for the trust-region subproblem, J. Optim. Theory Appl., 154 (2012), 904–915. https://doi.org/10.1007/s10957-012-0041-6 doi: 10.1007/s10957-012-0041-6
[29]	H. A. Le Thi, V. N. Huynh, T. P. Dinh, D.C. programming and DCA for general D.C. programs, in: T. Van Do, H. A. Le Thi, N. T. Nguyen (eds.), Advanced Computational Methods for Knowledge Engineering, Springer, Cham, 2014, 15–35.
[30]	H. Tuy, D.C. optimization: theory, methods and algorithms, in: R. Horst, P. M. Pardalos (eds.), Handbook of Global Optimization, Kluwer Academic Publisher, Dordrecht, 1995,149–216.
[31]	A. S. Strekalovsky, Global optimality conditions for optimal control problems with functions of A.D. Alexandrov, J. Optim. Theory Appl., 159 (2013), 297–321. https://doi.org/10.1007/s10957-013-0355-z doi: 10.1007/s10957-013-0355-z
[32]	A. S. Strekalovsky, On solving optimization problems with hidden nonconvex structures, in: T. M. Rassias, C. A. Floudas, S. Butenko (eds.), Optimization in Science and Engineering, Springer, New York, 2014,465–502. https://doi.org/10.1007/978-1-4939-0808-0_23
[33]	A. S. Strekalovsky, On local search in D.C. optimization problems, Appl. Math. Comput., 255 (2015), 73–83. https://doi.org/10.1016/j.amc.2014.08.092 doi: 10.1016/j.amc.2014.08.092
[34]	A. S. Strekalovsky, Global optimality conditions in nonconvex optimization, J. Optim. Theory Appl., 173 (2017), 770–792. https://doi.org/10.1007/s10957-016-0998-7 doi: 10.1007/s10957-016-0998-7
[35]	A. S. Strekalovsky, Global optimality conditions and exact penalization, Optim. Lett., 13 (2017), 597–615. https://doi.org/10.1007/s11590-017-1214-x doi: 10.1007/s11590-017-1214-x
[36]	A. S. Strekalovsky, I. M. Minarchenko, A local search method for optimization problem with D.C. inequality constraints, Appl. Math. Model., 58 (2018), 229–244. https://doi.org/10.1016/j.apm.2017.07.031 doi: 10.1016/j.apm.2017.07.031
[37]	A. Strekalovskiy, Nonconvex optimization: from global optimality conditions to numerical methods, in: Proceedings of the 14th International Global Optimization Workshop (LeGO 2018), Leiden, Netherlands, 2019, 20–70. https://doi.org/10.1063/1.5089982
[38]	A. S. Strekalovsky, New methods for solving nonconvex problems of optimization and optimal control, Paper presented at the International Conference Optimization and Applications (OPTIMA2009), Petrovac, Montenegro, September 21–25, 2009.
[39]	A. S. Strekalovsky, New global optimality conditions in a problem with D.C. constraints, Trudy Inst. Mat. i Mekh. UrO RAN, 25 (2019), 245–261 (in Russian).
[40]	A. S. Strekalovsky, Local search for nonsmooth D.C. optimization with D.C. equality and inequality constraints, in: A. M. Bagirov, M. Gaudioso, N. Karmitsa, M. M. Makela, S. Taheri (eds.), Numerical Nonsmooth Optimization, Springer, Cham, 2020,229–261. https://doi.org/10.1007/978-3-030-34910-3_7
[41]	A. S. Strekalovsky, On a global search in D.C. optimization problems, in: M. Jacimovic, M. Khachay, V. Malkova, M. Posypkin (eds.), OPTIMA 2019, CCIS, Springer, Cham, 2020,222–236. https://doi.org/10.1007/978-3-030-38603-0_17
[42]	F. P. Vasiliev, Methody Optimizatsii [Optimization Method], Factorial Press, Moscow, 2002 (in Russian).

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