Mathematical Programming

, Volume 165, Issue 1, pp 71–111 | Cite as

Two-stage stochastic variational inequalities: an ERM-solution procedure

  • Xiaojun ChenEmail author
  • Ting Kei Pong
  • Roger J-B. Wets
Full Length Paper Series B


We propose a two-stage stochastic variational inequality model to deal with random variables in variational inequalities, and formulate this model as a two-stage stochastic programming with recourse by using an expected residual minimization solution procedure. The solvability, differentiability and convexity of the two-stage stochastic programming and the convergence of its sample average approximation are established. Examples of this model are given, including the optimality conditions for stochastic programs, a Walras equilibrium problem and Wardrop flow equilibrium. We also formulate stochastic traffic assignments on arcs flow as a two-stage stochastic variational inequality based on Wardrop flow equilibrium and present numerical results of the Douglas–Rachford splitting method for the corresponding two-stage stochastic programming with recourse.


Stochastic variational inequalities Stochastic program with recourse Wardrop equilibrium Expected residual minimization Regularized gap function Splitting method 

Mathematics Subject Classification

90C33 90C15 


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  • Xiaojun Chen
    • 1
    Email author
  • Ting Kei Pong
    • 1
  • Roger J-B. Wets
    • 2
  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  2. 2.Department of MathematicsUniversity of California, DavisDavisUSA

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