Variational Convergence of Bifunctions: Motivating Applications

It is shown that a number of variational and equilibrium problems can be cast as finding the maxinf-points or minsup-points of bivariate functions, for short, bifunctions. These problems include linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, noncooperative games, and Walras and Nash equilibrium problems. One appeals to the theory of lopsided convergence for bifunctions to derive stability results for each one of these problems.

[1]  Gül Gürkan,et al.  Sample-path solution of stochastic variational inequalities , 1999, Math. Program..

[2]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[3]  Christian Kanzow,et al.  Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions , 2009, Comput. Optim. Appl..

[4]  Stephen M. Robinson,et al.  Analysis of Sample-Path Optimization , 1996, Math. Oper. Res..

[5]  Xiaojun Chen,et al.  Stochastic Variational Inequalities: Residual Minimization Smoothing Sample Average Approximations , 2012, SIAM J. Optim..

[6]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[7]  Jong-Shi Pang,et al.  Joint Sensing and Power Allocation in Nonconvex Cognitive Radio Games: Nash Equilibria and Distributed Algorithms , 2012, IEEE Transactions on Information Theory.

[8]  Fabián Flores Bazán,et al.  The Linear Complementarity Problem Under Asymptotic Analysis , 2005, Math. Oper. Res..

[9]  Jong-Shi Pang,et al.  The basic theorem of complementarity revisited , 1993, Math. Program..

[10]  I. Konnov.  Equilibrium Models and Variational Inequalities , 2013 .

[11]  J. Danskin.  The Theory of Max-Min and its Application to Weapons Allocation Problems , 1967 .

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

[13]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[14]  Alejandro Jofré,et al.  Continuity Properties of Walras Equilibrium Points , 2002, Ann. Oper. Res..

[15]  Jong-Shi Pang,et al.  Joint Sensing and Power Allocation in Nonconvex Cognitive Radio Games: Quasi-Nash Equilibria , 2011, IEEE Transactions on Signal Processing.

[16]  Roger J.-B. Wets,et al.  Outer Semicontinuity of Positive Hull Mappings with Application to Semi-Infinite and Stochastic Programming , 2008, SIAM J. Optim..

[17]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[18]  Xiaojun Chen,et al.  On Smoothing Methods for the P[sub 0] Matrix Linear Complementarity Problem , 2000, SIAM J. Optim..

[19]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

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

[21]  Roger J.-B. Wets,et al.  Continuity of some convex-cone-valued mappings , 1967 .

[22]  Roger J.-B. Wets.  Lipschitz Continuity of inf-Projections , 2003, Comput. Optim. Appl..

[23]  Sjur Didrik Flåm,et al.  Equilibrium programming using proximal-like algorithms , 1997, Math. Program..

[24]  Luiz Carlos Matioli,et al.  A numerical algorithm for finding solutions of a generalized Nash equilibrium problem , 2012, Comput. Optim. Appl..

[25]  Yonina C. Eldar,et al.  Convex Optimization in Signal Processing and Communications , 2009 .

[26]  Hassan Riahi,et al.  Sensitivity analysis for abstract equilibrium problems , 2005 .

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

[28]  E. Cavazzuti.  Convergence of equilibria in the theory of games , 1986 .

[29]  R. Wets,et al.  A convergence theory for saddle functions , 1983 .

[30]  G. Debreu,et al.  Theory of Value , 1959 .

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

[32]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[33]  Christian Kanzow,et al.  A New Regularization Method for Mathematical Programs with Complementarity Constraints with Strong Convergence Properties , 2013, SIAM J. Optim..

[34]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[35]  Adib Bagh,et al.  Epi/Hypo–Convergence: The Slice Topology and Saddle Points Approximation , 1996 .

[36]  Stable approximations of set-valued maps , 1988 .

[37]  Roger J.-B. Wets,et al.  On the continuity of the value of a linear program and of related polyhedral-valued multifunctions , 1982 .