edocServer der HumboldtUniversität zu Berlin 
Author(s):  René Henrion, Weierstrass Institute 
Title:  Perturbation ananlysis of chanceconstrained programs under variation of all constraint data 
Date of Acceptance:  10.02.2003 
Submission Date:  09.01.2003 
Series Title: 
Stochastic Programming EPrint Series (SPEPS) 
Editors:  Julie L. Higle; Werner Römisch; Surrajeet Sen 
Appeared in: 
Lecture notes in economics and mathematical systems (Vol. 532, 2004)
Springer (Berlin) 
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Abstract (eng):  
A fairly general shape of chance constraint programs is \[ (P) min \{ g(x)  x \in X, \mu (H(x)) \le p \} , \] where $g : \R^m \to \R$ is a continuous objective function, $X \subseteq \R^m$ is a closed subset of deterministic constraints, and the inequality defines a probabilistic constraint with $H : \R^m \to \atop \to \R^s$ being a multifunction with closed graph, $\mu$ is a probability measure on $\rR^s$ and $p \in (0,1)$ is some probability level. In the simplest case of linear chance constraints, $g$ is linear, $X$ is a polyhedron and $H(x) = \{ z \in \R^s  Ax \ge z\} $, where $A$ is a matrix of order $(s,m)$ and the inequality sign has to be understood componentwise.\\ Since the data of optimization problems are typically uncertain or approximated by other data which are easier to handle, the question of stability of solutions arises naturally. Concerning $(P)$, the first idea is to investigate solutions under perturbations of the right hand side $p$ of the inequality. This reflects the modeling degree of freedom when choosing a probability at which the constraint system is supposed to be valid. Furthermore, the probability measure $\mu$ is unknown in general and has to be approximated, for instance, by empirical measures. This motivates to extend the perturbation analysis to $\mu$. Stability of solutions of $(P)$ with respect to $p$ and $\mu$ is well understood now but shall be briefly reviewed in this paper for the sake of being selfcontained. Apart from these two constraint parameters, also approximations of the deterministic constraint $X$ and of the random set mapping $H$ in $(P)$ may be of interest. The aim of this paper is to identify constraint qualifications for stability under partial p erturbations of the single constraint parameters in $(P)$. Due to the increasing complexity of how these parameters influence each other, the resulting constraint qualifications become more and more restrictive when passing from $p$ over $\mu$ to $X$ and $H$. Part of the result relate to convex data in $(P)$ or even in the perturbations of $(P)$. Special emphasis is put on a series of counterexamples highlighting the necessity and limitations of the obtained conditions.  
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