Discrete Approximation of Two-Stage Stochastic and Distributionally Robust Linear Complementarity Problems

TL;DR

This paper introduces a discretization scheme for two-stage stochastic and distributionally robust linear complementarity problems, providing convergence analysis and solution methods, with applications to duopoly game equilibrium modeling.

Contribution

It presents a novel discretization approach for two-stage stochastic and distributionally robust LCPs, including convergence results and solution techniques like progressive hedging.

Findings

Convergence of discretized solutions under moderate conditions.

Application of the method to a duopoly game equilibrium.

Extension to distributionally robust LCP with moment constraints.

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.