Necessary optimality conditions for twostage stochastic programs with equilibrium constraints
Xu, Huifu and Ye, Jane J. (2010) Necessary optimality conditions for twostage stochastic programs with equilibrium constraints. SIAM Journal on Optimization, 20, (4), 16851715. (doi:10.1137/090748974).
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Description/Abstract
Developing first order optimality conditions for twostage stochastic mathematical programs with equilibrium constraints (SMPECs) whose second stage problem has multiple equilibria/solutions is a challenging undone work. In this paper we take this challenge by considering a general class of twostage SMPECs whose equilibrium constraints are represented by a parametric variational inequality (where the first stage decision vector and a random vector are treated as parameters). We use the sensitivity analysis on deterministic mathematical programs with equilibrium constraints (MPECs) as a tool to deal with the challenge: First, we extend a wellknown theorem in nonsmooth analysis about the exchange of the subdifferential operator with Aumann's integration from a nonatomic probability space to a general setting; second, we apply the extended result together with the existing sensitivity analysis results on the value function of the deterministic MPEC and the bilevel programming to the value function of our second stage problem; third, we develop various optimality conditions in terms of the subdifferential of the value function of the second stage problem and its relaxations which are constructed through the gradients of the underlying function at the second stage; finally we analyze special cases when the variational inequality constraint reduces to a complementarity problem and further to a system of nonlinear equalities and inequalities. The subdifferential to be used in this paper is the limiting (Mordukhovich) subdifferential, and the probability space is not necessarily nonatomic which means that Aumann's integral of the limiting subdifferential of a random function may be strictly smaller than that of Clarke's.
Item Type:  Article  

Digital Object Identifier (DOI):  doi:10.1137/090748974  
ISSNs:  10526234 (print) 

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Keywords:  stochastic mathematical program with equilibrium constraints, first order necessary conditions, limiting subdifferentials, Mstationary points, random setvalued mappings, sensitivity analysis  
Subjects:  Q Science > Q Science (General)  
Divisions:  University Structure  Pre August 2011 > School of Mathematics > Operational Research 

ePrint ID:  79532  
Date : 


Date Deposited:  16 Mar 2010  
Last Modified:  31 Mar 2016 13:15  
URI:  http://eprints.soton.ac.uk/id/eprint/79532 
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