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
395-418
Shapiro, Alexander
c17bba20-6f82-47be-b8a3-859504691942
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
January 2008
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), .
(doi:10.1080/02331930801954177).
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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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
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Date deposited: 17 Mar 2010
Last modified: 14 Mar 2024 02:47
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Author:
Alexander Shapiro
Author:
Huifu Xu
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