Stochastic Nash equilibrium problems: sample average approximation and applications

Authors:
Huifu Xu;Dali Zhang
Affiliations:
School of Engineering and Mathematical Sciences, City University of London, London, UK EC1V 0HB;Sino-US Global Logistics Institute, Shanghai Jiao Tong University, Shanghai, China 200030
Venue:
Computational Optimization and Applications
Year:
2013

Citing 20
Cited 1

A linearization method for nonsmooth stochastic programming problems

Mathematics of Operations Research
Stochastic equilibrium programming for dynamic oligopolistic markets

Journal of Optimization Theory and Applications
Sensitivity analysis for nonsmooth generalized equations

Mathematical Programming: Series A and B
Quantitative stability in stochastic programming

Mathematical Programming: Series A and B
Analysis of sample-path optimization

Mathematics of Operations Research
Convergence properties of two-stage stochastic programming

Journal of Optimization Theory and Applications
Sensitivity Analysis of the Value Function for Optimization Problems with Variational Inequality Constraints

SIAM Journal on Control and Optimization
On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs

SIAM Journal on Optimization
Spatial Oligopolistic Electricity Models with Cournot Generators and Regulated Transmission Prices

Operations Research
Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

Mathematical Programming: Series A and B - Nonlinear convex optimization and variational inequalities
Generalized Nash equilibrium problems and Newton methods

Mathematical Programming: Series A and B
On Rates of Convergence for Stochastic Optimization Problems Under Non-Independent and Identically Distributed Sampling

SIAM Journal on Optimization
Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints

Mathematics of Operations Research
Using EPECs to Model Bilevel Games in Restructured Electricity Markets with Locational Prices

Operations Research
Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network

Operations Research
Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Mathematical Programming: Series A and B
A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application

Operations Research
Probabilistic analysis of simulation-based games

ACM Transactions on Modeling and Computer Simulation (TOMACS)
Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach

Mathematics of Operations Research
Game Theoretic Cross-Layer Transmission Policies in Multipacket Reception Wireless Networks

IEEE Transactions on Signal Processing

Sample average approximation method for a class of stochastic generalized Nash equilibrium problems

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.00

Visualization

Abstract

This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The well-known sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a statistical estimator (a Nash equilibrium or a Nash-C-stationary point) obtained from sample average approximate equilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the Clarke generalized derivatives, it is shown that with probability approaching one exponentially fast by increasing sample size, the Nash-C-stationary point converges to a weak Nash-C-stationary point of the true problem. Finally, the model is applied to stochastic Nash equilibrium problem in the wholesale electricity market.