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April  2018, 14(2): 637-646. doi: 10.3934/jimo.2017065

Solving normalized stationary points of a class of equilibrium problem with equilibrium constraints

School of Sciences, Southwest Petroleum University, Xindu Avenue No.8, Xindu District, Chengdu City, 610500, China

* Corresponding author: Peiyu li

Received  March 2016 Published  April 2018 Early access  June 2017

Fund Project: The author is supported by NSFC grant 11501476.

This paper focuses on solving normalized stationary points of a class of equilibrium problem with equilibrium constraints (EPEC). We show that, under some kind of separability assumption, normalized C-/M-/S-stationary points of EPEC are actually C-/M-/S-stationary points of an associated mathematical program with equilibrium constraints (MPEC), which implies that we can solve MPEC to obtain normalized stationary points of EPEC. In addition, we demonstrate the proposed approach on competition of manufacturers for similar products in the same city.

Citation: Peiyu Li. Solving normalized stationary points of a class of equilibrium problem with equilibrium constraints. Journal of Industrial and Management Optimization, 2018, 14 (2) : 637-646. doi: 10.3934/jimo.2017065
References:
[1]

F. FacchineiA. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Operations Research Letters, 35 (2007), 159-164.  doi: 10.1016/j.orl.2006.03.004.

[2]

J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23-39.  doi: 10.1007/s00607-004-0083-1.

[3]

M. L. Flegel and C. Kanzow, On M-stationary points for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 310 (2005), 286-302.  doi: 10.1016/j.jmaa.2005.02.011.

[4]

R. Fletcher and S. Leyffer, Solving mathematical programs with complementarity constraints as nonlinear programs, Optimization Methods and Software, 19 (2004), 15-40.  doi: 10.1080/10556780410001654241.

[5]

R. FletcherS. LeyfferD. Ralph and S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 17 (2006), 259-286.  doi: 10.1137/S1052623402407382.

[6]

L. Guo and G. H. Lin, Global algorithm for solving stationary points for equilibrium programs with shared equilibrium constraints, Pacific Journal of Optimization, 9 (2013), 443-461. 

[7]

L. Guo and G. H. Lin, Notes on some constraint qualifications for mathematical programs with equilibrium constraints, Journal of Optimization Theory and Applications, 156 (2013), 600-616.  doi: 10.1007/s10957-012-0084-8.

[8]

L. GuoG. H. LinD. Zhang and D. Zhu, An MPEC reformulation of an EPEC model for electricity markets, Operations Research Letters, 43 (2015), 262-267.  doi: 10.1016/j.orl.2015.03.001.

[9]

P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operations Research, 54 (1991), 81-94.  doi: 10.1016/0377-2217(91)90325-P.

[10]

M. Hu and M. Fukushima, Variational inequality formulation of a class of multi-leader-follower games, Journal of Optimization Theory and Applications, 151 (2011), 455-473.  doi: 10.1007/s10957-011-9901-8.

[11]

M. Hu and M. Fukushima, Existence, uniqueness, and computation of robust Nash equilibrium in a class of multi-leader-follower games, SIAM Journal on Optimization, 23 (2013), 894-916.  doi: 10.1137/120863873.

[12]

X. Hu, Mathematical Programs with Complementarity Constraints and Game Theory Models in Electricity Markets, Ph. D thesis, Department of Mathematics and Statistics, University of Melbourne, 2003.

[13]

A. A. Kulkarni and U. V. Shanbhag, A shared-constraint approach to multi-leader multi-follower games, Set-Valued and Variational Analysis, 22 (2014), 691-720.  doi: 10.1007/s11228-014-0292-5.

[14]

A. A. Kulkarni and U. V. Shanbhag, On the consistency of leaders' conjectures in hierarchical games, 52nd IEEE Annual Conference on Decision and Control (CDC), (2013), 1180-1185.  doi: 10.1109/CDC.2013.6760042.

[15]

S. Leyffer and T. Munson, Solving multi-leader-common-follower games, Optimization Methods and Software, 25 (2010), 601-623.  doi: 10.1080/10556780903448052.

[16]

Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996. doi: 10.1017/CBO9780511983658.

[17]

B. S. Mordukhovich, Optimization and equilibrium problems with equilibrium constraints in infinite-dimensional spaces, Optimization, 57 (2008), 715-741.  doi: 10.1080/02331930802355390.

[18]

J. V. Outrata, A note on a class of equilibrium problems with equilibrium constraints, Kybernetika, 40 (2004), 585-594. 

[19]

J. S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Computational Management Science, 2 (2005), 21-56.  doi: 10.1007/s10287-004-0010-0.

[20]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.

[21]

C. L. Su, A sequential NCP algorithm for solving equilibrium problems with equilibrium constraints, Technical Report, Department of Management Science and Engineering, Stanford University, 2004.

[22]

H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity, Mathematics of Operations Research, 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213.

[23]

S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11 (2001), 918-936.  doi: 10.1137/S1052623499361233.

[24]

J. J. Ye, Optimality conditions for optimization problems with complementarity constraints, SIAM Journal on Optimization, 9 (1999), 374-387.  doi: 10.1137/S1052623497321882.

[25]

J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 307 (2005), 350-369.  doi: 10.1016/j.jmaa.2004.10.032.

show all references

References:
[1]

F. FacchineiA. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Operations Research Letters, 35 (2007), 159-164.  doi: 10.1016/j.orl.2006.03.004.

[2]

J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23-39.  doi: 10.1007/s00607-004-0083-1.

[3]

M. L. Flegel and C. Kanzow, On M-stationary points for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 310 (2005), 286-302.  doi: 10.1016/j.jmaa.2005.02.011.

[4]

R. Fletcher and S. Leyffer, Solving mathematical programs with complementarity constraints as nonlinear programs, Optimization Methods and Software, 19 (2004), 15-40.  doi: 10.1080/10556780410001654241.

[5]

R. FletcherS. LeyfferD. Ralph and S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 17 (2006), 259-286.  doi: 10.1137/S1052623402407382.

[6]

L. Guo and G. H. Lin, Global algorithm for solving stationary points for equilibrium programs with shared equilibrium constraints, Pacific Journal of Optimization, 9 (2013), 443-461. 

[7]

L. Guo and G. H. Lin, Notes on some constraint qualifications for mathematical programs with equilibrium constraints, Journal of Optimization Theory and Applications, 156 (2013), 600-616.  doi: 10.1007/s10957-012-0084-8.

[8]

L. GuoG. H. LinD. Zhang and D. Zhu, An MPEC reformulation of an EPEC model for electricity markets, Operations Research Letters, 43 (2015), 262-267.  doi: 10.1016/j.orl.2015.03.001.

[9]

P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operations Research, 54 (1991), 81-94.  doi: 10.1016/0377-2217(91)90325-P.

[10]

M. Hu and M. Fukushima, Variational inequality formulation of a class of multi-leader-follower games, Journal of Optimization Theory and Applications, 151 (2011), 455-473.  doi: 10.1007/s10957-011-9901-8.

[11]

M. Hu and M. Fukushima, Existence, uniqueness, and computation of robust Nash equilibrium in a class of multi-leader-follower games, SIAM Journal on Optimization, 23 (2013), 894-916.  doi: 10.1137/120863873.

[12]

X. Hu, Mathematical Programs with Complementarity Constraints and Game Theory Models in Electricity Markets, Ph. D thesis, Department of Mathematics and Statistics, University of Melbourne, 2003.

[13]

A. A. Kulkarni and U. V. Shanbhag, A shared-constraint approach to multi-leader multi-follower games, Set-Valued and Variational Analysis, 22 (2014), 691-720.  doi: 10.1007/s11228-014-0292-5.

[14]

A. A. Kulkarni and U. V. Shanbhag, On the consistency of leaders' conjectures in hierarchical games, 52nd IEEE Annual Conference on Decision and Control (CDC), (2013), 1180-1185.  doi: 10.1109/CDC.2013.6760042.

[15]

S. Leyffer and T. Munson, Solving multi-leader-common-follower games, Optimization Methods and Software, 25 (2010), 601-623.  doi: 10.1080/10556780903448052.

[16]

Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996. doi: 10.1017/CBO9780511983658.

[17]

B. S. Mordukhovich, Optimization and equilibrium problems with equilibrium constraints in infinite-dimensional spaces, Optimization, 57 (2008), 715-741.  doi: 10.1080/02331930802355390.

[18]

J. V. Outrata, A note on a class of equilibrium problems with equilibrium constraints, Kybernetika, 40 (2004), 585-594. 

[19]

J. S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Computational Management Science, 2 (2005), 21-56.  doi: 10.1007/s10287-004-0010-0.

[20]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, 33 (1965), 520-534.  doi: 10.2307/1911749.

[21]

C. L. Su, A sequential NCP algorithm for solving equilibrium problems with equilibrium constraints, Technical Report, Department of Management Science and Engineering, Stanford University, 2004.

[22]

H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity, Mathematics of Operations Research, 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213.

[23]

S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11 (2001), 918-936.  doi: 10.1137/S1052623499361233.

[24]

J. J. Ye, Optimality conditions for optimization problems with complementarity constraints, SIAM Journal on Optimization, 9 (1999), 374-387.  doi: 10.1137/S1052623497321882.

[25]

J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, Journal of Mathematical Analysis and Applications, 307 (2005), 350-369.  doi: 10.1016/j.jmaa.2004.10.032.

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