Huifu Xu

Two stage stochastic equilibrium problems with equilibrium constraints: modeling and numerical schemes

  • Huifu Xu
  • Dali Zhang
    • Categories Complementarity and Variational Inequalities, Game Theory, Stochastic Programming Tags , , ,

    This paper presents a two stage stochastic equilibrium problem with equilibrium constraints(SEPEC) model. Some source problems which motivate the model are discussed. Monte Carlo sampling method is applied to solve the SEPEC. The convergence analysis on the statistical estimators of Nash equilibria and Nash stationary points are presented. Article Download View Two stage stochastic equilibrium … Read more

    Approximating Stationary Points of Stochastic Mathematical Programs with Equilibrium Constraints via Sample Averaging

  • Huifu Xu
  • Jane J. Ye
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization, Stochastic Programming Tags , , , , ,

    We investigate sample average approximation of a general class of one-stage stochastic mathematical programs with equilibrium constraints. By using graphical convergence of unbounded set-valued mappings, we demonstrate almost sure convergence of a sequence of stationary points of sample average approximation problems to their true counterparts as the sample size increases. In particular we show the … Read more

    Stability Analysis of Two Stage Stochastic Mathematical Programs with Complementarity Constraints via NLP-Regularization

  • Gui-Hua Lin
  • Yongchao Liu
  • Huifu Xu
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , ,

    This paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint: first, the complementarity constraint is approximated by a parameterized system of inequalities with a well-known regularization approach (SIOPT, Vol.11, 918-936) in deterministic mathematical programs with equilibrium constraints; the distribution of the … Read more

    Necessary Optimality Conditions for two-stage Stochastic Programs with Equilibrium Constraints

  • Huifu Xu
  • Jane J. Ye
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization, Stochastic Programming Tags , , , , ,

    Developing first order optimality conditions for a two-stage stochastic mathematical program with equilibrium constraints (SMPEC) whose second stage problem has multiple equilibria/solutions is a challenging undone work. In this paper we take this challenge by considering a general class of two-stage whose equilibrium constraints are represented by a parametric variational inequality (where the first stage … Read more

    Stochastic Nash Equilibrium Problems: Sample Average Approximation and Applications

  • Huifu Xu
  • Dali Zhang
    • Categories Game Theory, Nonsmooth Optimization, Stochastic Programming Tags , , ,

    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 … Read more

    A Two Stage Stochastic Equilibrium Model for Electricity Markets with Two Way Contracts

  • Yue Wu
  • Huifu Xu
  • Dali Zhang
    • Categories Applications - OR and Management Sciences, Complementarity and Variational Inequalities

    This paper investigates generators’ strategic behaviors in contract signing in the forward market and power transaction in the electricity spot market. A stochastic equilibrium program with equilibrium constraints (SEPEC) model is proposed to characterize the interaction of generators’ competition in the two markets. The model is an extension of a similar model proposed by Gans, … Read more

    Uniform Laws of Large Numbers for Set-Valued Mappings and Subdifferentials of Random Functions

  • Alexander Shapiro
  • Huifu Xu
    • Categories Stochastic Programming Tags , , , , , ,

    We derive a uniform (strong) Law of Large Numbers (LLN) for random set-valued mappings. The result can be viewed as an extension of both, a uniform LLN for random functions and LLN for random sets. We apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs. … Read more

    Stochastic Mathematical Programs with Equilibrium Constraints, Modeling and Sample Average Approximation

  • Alexander Shapiro
  • Huifu Xu
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , , ,

    In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate piecewise structure and directional differentiability of both — the lower level equilibrium solution and objective integrant. We show almost sure convergence of optimal values, optimal solutions … Read more