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

Generalized Nash equilibrium problem based on malfatti's problem

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.

Citation: Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
References:
[1]

Ma rco AndreattaAn 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. EnkhbatM. 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 FischerMarkus 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.

show all references

References:
[1]

Ma rco AndreattaAn 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. EnkhbatM. 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 FischerMarkus 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.

Figure 1.  Stationary Nash equilibrium
Figure 2.  Case 1
Figure 3.  Case 2
Figure 4.  Case 1
Figure 5.  Case 2
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