Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems

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Abstract

In this paper, we propose a discretization scheme for the two-stage stochastic linear complementarity problem (LCP) where the underlying random data are continuously distributed. Under some moderate conditions, we derive qualitative and quantitative convergence for the solutions obtained from solving the discretized two-stage stochastic LCP (SLCP). We explain how the discretized two-stage SLCP may be solved by the well-known progressive hedging method (PHM). Moreover, we extend the discussion by considering a two-stage distributionally robust LCP (DRLCP) with moment constraints and proposing a discretization scheme for the DRLCP. As an application, we show how the SLCP and DRLCP models can be used to study equilibrium arising from two-stage duopoly game where each player plans to set up its optimal capacity at present with anticipated competition for production in future.

Keywords

Two-stage stochastic linear complementarity problem Discrete approximation Error bound Distributionally robust linear complementarity problem Ex post equilibrium 

Mathematics Subject Classification

90C15 90C33 65K15 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongHong Kong
  2. 2.School of Economics and ManagementNanjing University of Science and TechnologyNanjingChina
  3. 3.School of MathematicsUniversity of SouthamptonSouthamptonUK

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