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Stationary Nash equilibrium
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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 |
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.
Mathematics Subject Classification:
Primary: 80M50; Secondary: 65K10.
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
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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. |
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.
|
| [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. |
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