Generalized Nash equilibrium problem based on malfatti's problem

Enkhbat Rentsen; Battur Gompil

doi:10.3934/naco.2020022

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2021, Volume 11, Issue 2: 209-220. Doi: 10.3934/naco.2020022

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Generalized Nash equilibrium problem based on malfatti's problem

Enkhbat Rentsen^1, and
Battur Gompil^2, ,

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

^* Corresponding author: Battur Gompil
^* Corresponding author: Battur Gompil

Received: August 2019

Revised: December 2019

Early access: March 2020

Published: June 2021

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Abstract

In this paper we consider non-cooperative game problem based on the Malfatti's problem. This problem is a special case of generalized Nash equilibrium problems with nonconvex shared constraints. Some numerical results are provided.

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Mathematics Subject Classification: Primary: 80M50; Secondary: 65K10.

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Figure 1. Stationary Nash equilibrium

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Figure 2. Case 1

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Figure 3. Case 2

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Figure 4. Case 1

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Figure 5. Case 2

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References

[1]	Ma rco Andreatta, An drás Bezdek and Jan P. Boroński, The problem of Malfatti: Two centuries of debate, The Mathematical Intelligencer, 33 (2011), 72-76.
[2]	G. Debreu, A social equilibrium existence theorem, Proceedings of the National Academy of Sciencesof the United States of America, 38 (1952), 886-893. doi: 10.1073/pnas.38.10.886.
[3]	R. Enkhbat, An algorithm for maximizing a convex function over a simple set, Journal of Global Optimization, 8 (1996), 379-391. doi: 10.1007/BF02403999.
[4]	R. Enkhbat, Global optimization approach to Malfatti's problem, Journal of Global Optimization, 65 (2016), 3-39. doi: 10.1007/s10898-015-0372-6.
[5]	R. Enkhbat, M. V. Barkova and A. S. Strekalovsky, Solving Malfatti's high dimensional problem by global optimization, Numerical Algebra, Control and Optimization, 2 (2016), 153-160. doi: 10.3934/naco.2016005.
[6]	F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, Annals of Operations Research, 1 (2010), 177-211. doi: 10.1007/s10479-009-0653-x.
[7]	Andreas Fischer, Markus Herrich and Klaus Schonefeld, Generalized Nash equilibrium problems - Recent advances and challenges, Pesquisa Operacional, 3 (2014), 521-558.
[8]	M. Fukushima, Restricted generalized Nash equilibria and controlled penalty algorithm, Technical Report, Department of Applied Mathematics and Physics, Kyoto University, 2008-007, July (2008). doi: 10.1007/s10287-009-0093-8.
[9]	H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 5 (1967).
[10]	M. Goldberg, On the original Malfatti problem, Math. Mag., 5 (1967), 241-247.
[11]	A. Heusinger and C. Kanzow, Relaxation methods for generalized Nash equilibrium problems with inexact line search, Journal of Optimization Theory and Applications, 1 (2009), 159-183. doi: 10.1007/s10957-009-9553-0.
[12]	K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem, Journal of Optimization Theory and Applications, 3 (2010), 511-531. doi: 10.1007/s10957-009-9614-4.
[13]	H. Lob and H. W. Richmond, On the solutions of the Malfatti problem for a triangle, Proc. London Math. Soc., 30 (1930), 287-301. doi: 10.1112/plms/s2-30.1.287.
[14]	G. A. Los, Malfatti's Optimization Problem, Dep. Ukr. NIINTI July 5, [in Russian], 1988.
[15]	C. Malfatti, Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa ttaliana della Scienze, 1 (1803), 235-244.
[16]	A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066.
[17]	J.-Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112.
[18]	V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25.
[19]	V. A. Zalgaller and G. A. Los, The solution of Malfatti's problem, Journal of Mathematical Sciences, 4 (1994), 3163-3177. doi: 10.1007/BF01249514.