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Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation

Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation
Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation
In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both-the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush-Kuhn-Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results
stochastic programming, equilibrium constraints, stackelberg-nash-cournot equilibrium, variational inequality, sample average approximation, exponential convergence, smoothing
0233-1934
395-418
Shapiro, Alexander
c17bba20-6f82-47be-b8a3-859504691942
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Shapiro, Alexander
c17bba20-6f82-47be-b8a3-859504691942
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5

Shapiro, Alexander and Xu, Huifu (2008) Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation. Optimization, 57 (3), 395-418. (doi:10.1080/02331930801954177).

Record type: Article

Abstract

In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both-the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush-Kuhn-Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results

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More information

Published date: January 2008
Keywords: stochastic programming, equilibrium constraints, stackelberg-nash-cournot equilibrium, variational inequality, sample average approximation, exponential convergence, smoothing
Organisations: Operational Research

Identifiers

Local EPrints ID: 79539
URI: http://eprints.soton.ac.uk/id/eprint/79539
ISSN: 0233-1934
PURE UUID: 8bc1bcfd-0603-4bd0-a750-22466a5feb18
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

Catalogue record

Date deposited: 17 Mar 2010
Last modified: 10 Dec 2019 01:46

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