Quantitative Stability Analysis of Stochastic Generalized Equations

We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibri...

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Bibliographic Details
Published in:SIAM journal on optimization Vol. 24; no. 1; pp. 467 - 497
Main Authors: Liu, Yongchao, Römisch, Werner, Xu, Huifu
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects:
Applied mathematics
Approximation
Continuity
Dominance
Equilibrium
Expected values
Functions (mathematics)
Mapping
Mathematical analysis
Optimization
Stability analysis
Stochasticity
ISSN:1052-6234, 1095-7189
Online Access:Get full text
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Summary:We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stochastic variational inequality, stationary points of classical one-stage and two-stage stochastic minimization problems, two-stage stochastic mathematical programs with equilibrium constraints, and stochastic programs with second order dominance constraints. [PUBLICATION ABSTRACT]
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ISSN:1052-6234
1095-7189
DOI:10.1137/120880434