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June  2021, 11(2): 209-220. doi: 10.3934/naco.2020022

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

Received  August 2019 Revised  December 2019 Published  June 2021 Early access  March 2020

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.

Keywords: Generalized Nash equilibrium problem, Malfatti's problem, non-zero sum game.
Mathematics Subject Classification: Primary: 80M50; Secondary: 65K10.
Citation: Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Andreas Fischer, Markus Herrich and Klaus Schonefeld, Generalized Nash equilibrium problems - Recent advances and challenges, Pesquisa Operacional, 3 (2014), 521-558.   Google Scholar

[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.  Google Scholar

[9]

H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 5 (1967).  Google Scholar

[10]

M. Goldberg, On the original Malfatti problem, Math. Mag., 5 (1967), 241-247.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. A. Los, Malfatti's Optimization Problem, Dep. Ukr. NIINTI July 5, [in Russian], 1988. Google Scholar

[15]

C. Malfatti, Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa ttaliana della Scienze, 1 (1803), 235-244.   Google Scholar

[16]

A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066.   Google Scholar

[17]

J.-Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112.   Google Scholar

[18]

V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25.   Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Andreas Fischer, Markus Herrich and Klaus Schonefeld, Generalized Nash equilibrium problems - Recent advances and challenges, Pesquisa Operacional, 3 (2014), 521-558.   Google Scholar

[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.  Google Scholar

[9]

H. Gabai and E. Liban, On Goldberg's inequality associated with the Malfatti problem, Math. Mag., 5 (1967).  Google Scholar

[10]

M. Goldberg, On the original Malfatti problem, Math. Mag., 5 (1967), 241-247.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. A. Los, Malfatti's Optimization Problem, Dep. Ukr. NIINTI July 5, [in Russian], 1988. Google Scholar

[15]

C. Malfatti, Memoria sopra una problema stereotomico, Memoria di Matematica e di Fisica della Societa ttaliana della Scienze, 1 (1803), 235-244.   Google Scholar

[16]

A. S. Strekalovsky, On the global extrema problem, Soviet Math. Doklad, 292 (1987), 1062-1066.   Google Scholar

[17]

J.-Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices, Oper. Res., 47 (1999), 102-112.   Google Scholar

[18]

V. A. Zalgaller, An inequality for acute triangles, Ukr. Geom. Sb., 34 (1991), 10-25.   Google Scholar

[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.  Google Scholar

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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