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Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

Abstract

The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation.

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Notes

  1. 1.

    The analog in numerical analysis lies in the theory of overdetermined systems of equations generated by noise in measurements of parameters.

  2. 2.

    In part, this has recently been taken up by Chen et al. in [18] in terms of discrete approximations to two-stage stochastic variational inequality problems based on “continuous” probability instead of “discrete” probability, as here.

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Correspondence to R. Tyrrell Rockafellar.

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Rockafellar, R.T., Sun, J. Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174, 453–471 (2019) doi:10.1007/s10107-018-1251-y

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Keywords

  • Progressive hedging algorithm
  • Stochastic variational inequality problems
  • Stochastic complementarity problems
  • Stochastic programming problems
  • Maximal monotone mappings
  • Proximal point algorithm
  • Problem decomposition

Mathematics Subject Classification

  • 90C15
  • 49J20
  • 47H05