The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

We consider linear generalized Nash games and introduce the so-called cone condition, which characterizes the smoothness of a gap function that arises from a reformulation of the generalized Nash equilibrium problem as a piecewise linear optimization problem based on the Nikaido–Isoda function. Other regularity conditions such as the linear independence constraint qualification or the strict Mangasarian–Fromovitz condition are only sufficient for smoothness, but have the advantage that they can be verified more easily than the cone condition. Therefore, we present special cases, where these conditions are not only sufficient, but also necessary for smoothness of the gap function. Our main tool in the analysis is a global extension of the gap function that allows us to overcome the common difficulty that its domain may not cover the whole space.

[1]  A. F. Izmailov,et al.  Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it , 2015 .

[2]  Oliver Stein,et al.  On differentiability properties of player convex generalized Nash equilibrium problems , 2015 .

[3]  Christian Kanzow,et al.  On a Smooth Dual Gap Function for a Class of Player Convex Generalized Nash Equilibrium Problems , 2015, J. Optim. Theory Appl..

[4]  Oliver Stein,et al.  Bi-Level Strategies in Semi-Infinite Programming , 2003 .

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

[6]  Francisco Facchinei,et al.  Generalized Nash Equilibrium Problems , 2010, Ann. Oper. Res..

[7]  Hubertus Th. Jongen,et al.  Nonlinear Optimization in Finite Dimensions - Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects , 2000 .

[8]  William W. Cooper,et al.  Introduction to Data Envelopment Analysis and Its Uses: With Dea-Solver Software and References , 2005 .

[9]  H. Nikaidô,et al.  Note on non-cooperative convex game , 1955 .

[10]  Gerd Wachsmuth,et al.  On LICQ and the uniqueness of Lagrange multipliers , 2013, Oper. Res. Lett..

[11]  Francisco Facchinei,et al.  On the solution of the KKT conditions of generalized Nash equilibrium problems , 2011, SIAM J. Optim..

[12]  G. Franco.  Separation of Sets and Gap Functions for Quasi-Variational Inequalities , 1995 .

[13]  Jong-Shi Pang,et al.  On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method , 2013, Math. Program..

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

[15]  Christian Kanzow,et al.  On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities , 2014, J. Optim. Theory Appl..

[16]  Nguyen Dong Yen,et al.  On the Problem of Minimizing a Difference of Polyhedral Convex Functions Under Linear Constraints , 2016, J. Optim. Theory Appl..

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

[18]  Axel Dreves.  Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets , 2014, Math. Methods Oper. Res..

[19]  Andreas Fischer,et al.  GENERALIZED NASH EQUILIBRIUM PROBLEMS - RECENT ADVANCES AND CHALLENGES , 2014 .

[20]  A. Auslender.  Optimisation : méthodes numériques , 1976 .

[21]  Oliver Stein,et al.  On smoothness properties of optimal value functions at the boundary of their domain under complete convexity , 2014, Math. Methods Oper. Res..

[22]  Kaoru Tone,et al.  Data Envelopment Analysis , 1996 .

[23]  Oliver Stein,et al.  The C-Index: A New Stability Concept for Quadratic Programs with Complementarity Constraints , 2011, Math. Oper. Res..

[24]  H. Jongen,et al.  Nonlinear Optimization in Finite Dimensions , 2001 .

[25]  Oliver Stein,et al.  Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems , 2012, J. Glob. Optim..

[26]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[27]  A. F. Izmailov,et al.  Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it , 2015 .

[28]  F. Giannessi,et al.  Variational inequalities and network equilibrium problems , 1995 .

[29]  Jerzy Kyparisis,et al.  On uniqueness of Kuhn-Tucker multipliers in nonlinear programming , 1985, Math. Program..

[30]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

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

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

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