Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players’ strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.

[1]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[2]  J. Nash,et al.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[3]  Heinrich von Stackelberg,et al.  Stackelberg (Heinrich von) - The Theory of the Market Economy, translated from the German and with an introduction by Alan T. PEACOCK. , 1953 .

[4]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[5]  Peter Kall,et al.  Approximation to Optimization Problems: An Elementary Review , 1986, Math. Oper. Res..

[6]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[7]  A. Fischer.  A special newton-type optimization method , 1992 .

[8]  Christian Kanzow,et al.  Some Noninterior Continuation Methods for Linear Complementarity Problems , 1996, SIAM J. Matrix Anal. Appl..

[9]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[10]  Masao Fukushima,et al.  Solving box constrained variational inequalities by using the natural residual with D-gap function globalization , 1998, Oper. Res. Lett..

[11]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[12]  Masao Fukushima,et al.  A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints , 1998, Comput. Optim. Appl..

[13]  Michel Théra,et al.  Ill-posed Variational Problems and Regularization Techniques , 1999 .

[14]  Francisco Facchinei,et al.  A smoothing method for mathematical programs with equilibrium constraints , 1999, Math. Program..

[15]  J. Pang,et al.  Convergence of a Smoothing Continuation Method for Mathematical Progams with Complementarity Constraints , 1999 .

[16]  J. Pang,et al.  Strategic gaming analysis for electric power systems: an MPEC approach , 2000 .

[17]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[18]  Xiaojun Chen,et al.  A Smoothing Method for a Mathematical Program with P-Matrix Linear Complementarity Constraints , 2004, Comput. Optim. Appl..

[19]  Daniel Ralph,et al.  Using EPECs to Model Bilevel Games in Restructured Electricity Markets with Locational Prices , 2007, Oper. Res..

[20]  Gül Gürkan,et al.  Approximations of Nash equilibria , 2008, Math. Program..

[21]  Masao Fukushima,et al.  Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games , 2009, Comput. Manag. Sci..

[22]  Roger J.-B. Wets,et al.  Variational convergence of bivariate functions: lopsided convergence , 2008, Math. Program..

[23]  Jean Bosco Etoa Etoa.  A globally convergent sequential linear programming algorithm for mathematical programs with linear complementarity constraints , 2010 .