Stochastic variational inequalities: Residual minimization smoothing sample average approximations

Xiaojun Chen, Roger J B Wets, Yanfang Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

57 Citations (Scopus)

Abstract

The stochastic variational inequality (VI) has been used widely in engineering and economics as an effective mathematical model for a number of equilibrium problems involving uncertain data. This paper presents a new expected residual minimization (ERM) formulation for a class of stochastic VI. The objective of the ERM-formulation is Lipschitz continuous and semismooth which helps us guarantee the existence of a solution and convergence of approximation methods. We propose a globally convergent (a.s.) smoothing sample average approximation (SSAA) method to minimize the residual function; this minimization problem is convex for the linear stochastic VI if the expected matrix is positive semidefinite. We show that the ERM problem and its SSAA problems have minimizers in a compact set and any cluster point of minimizers and stationary points of the SSAA problems is a minimizer and a stationary point of the ERM problem (a.s.). Our examples come from applications involving traffic flow problems. We show that the conditions we impose are satisfied and that the solutions, efficiently generated by the SSAA procedure, have desirable properties.
Original languageEnglish
Pages (from-to)649-673
Number of pages25
JournalSIAM Journal on Optimization
Volume22
Issue number2
DOIs
Publication statusPublished - 7 Sep 2012

Keywords

  • Epi-convergence
  • Expected residual minimization
  • Lower semicontinuous
  • Semismooth
  • Smoothing sample average approximation
  • Stationary point
  • Stochastic variational inequalities
  • Upper semicontinuous

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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