Convergence of stationary points of sample average two-stage stochastic programs: a generalized equation approach
Convergence of stationary points of sample average two-stage stochastic programs: a generalized equation approach
This paper presents an asymptotic analysis of a Monte Carlo method, variously known as sample average approximation (SAA) or sample path optimization (SPO), for a general two stage stochastic minimization problem. We study the case when the second stage problem may have multiple local optima or stationary points that are not global solutions, and SAA is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points. New optimality conditions are developed for both the true problem and its SAA problem to accommodate Karush-Kuhn-Tucker points. Since the optimality conditions are essentially stochastic generalized equations, the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions. It is shown, under moderate conditions, that with probability~one an accumulation point of the SAA stationary points satisfies a relaxed stationary condition for the true problem and, further, that with probability approaching one exponentially fast with increasing sample size, a stationary point of SAA converges to the set of relaxed stationary points. These results strengthen or complement existing results where the second stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an SAA problem rather than the other way round. The results obtained for generalized equations also contribute to recent theory for stochastic variational inequalities and stochastic generalized equations.
stochastic generalized equation, sample average approximation, stochastic piecewise continuous (pc0), set-valued mappings
568-592
Ralph, Daniel
cb3e5a5e-f719-4dfd-bbe1-976aff170691
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
August 2011
Ralph, Daniel
cb3e5a5e-f719-4dfd-bbe1-976aff170691
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Ralph, Daniel and Xu, Huifu
(2011)
Convergence of stationary points of sample average two-stage stochastic programs: a generalized equation approach.
Mathematics of Operations Research, 36 (3), .
(doi:10.1287/moor.1110.0506).
Abstract
This paper presents an asymptotic analysis of a Monte Carlo method, variously known as sample average approximation (SAA) or sample path optimization (SPO), for a general two stage stochastic minimization problem. We study the case when the second stage problem may have multiple local optima or stationary points that are not global solutions, and SAA is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points. New optimality conditions are developed for both the true problem and its SAA problem to accommodate Karush-Kuhn-Tucker points. Since the optimality conditions are essentially stochastic generalized equations, the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions. It is shown, under moderate conditions, that with probability~one an accumulation point of the SAA stationary points satisfies a relaxed stationary condition for the true problem and, further, that with probability approaching one exponentially fast with increasing sample size, a stationary point of SAA converges to the set of relaxed stationary points. These results strengthen or complement existing results where the second stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an SAA problem rather than the other way round. The results obtained for generalized equations also contribute to recent theory for stochastic variational inequalities and stochastic generalized equations.
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Published date: August 2011
Keywords:
stochastic generalized equation, sample average approximation, stochastic piecewise continuous (pc0), set-valued mappings
Organisations:
Operational Research
Identifiers
Local EPrints ID: 205077
URI: http://eprints.soton.ac.uk/id/eprint/205077
ISSN: 0364-765X
PURE UUID: 93fc93fc-bdb8-498c-b374-4a45ba29678a
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Date deposited: 05 Dec 2011 15:17
Last modified: 15 Mar 2024 03:15
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Author:
Daniel Ralph
Author:
Huifu Xu
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