2003-02-10Buch DOI: 10.18452/8285
Perturbation ananlysis of chance-constrained programs under variation of all constraint data
 dc.contributor.author Henrion, René dc.contributor.editor Higle, Julie L. dc.contributor.editor Römisch, Werner dc.contributor.editor Sen, Surrajeet dc.date.accessioned 2017-06-16T19:51:08Z dc.date.available 2017-06-16T19:51:08Z dc.date.created 2006-02-22 dc.date.issued 2003-02-10 dc.date.submitted 2003-01-09 dc.identifier.uri http://edoc.hu-berlin.de/18452/8937 dc.description.abstract 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 component-wise.\\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 counter-examples highlighting the necessity and limitations of the obtained conditions. eng dc.language.iso eng dc.publisher Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ dc.subject.ddc 510 Mathematik dc.title Perturbation ananlysis of chance-constrained programs under variation of all constraint data dc.type book dc.identifier.urn urn:nbn:de:kobv:11-110-18452/8937-7 dc.identifier.doi http://dx.doi.org/10.18452/8285 local.edoc.container-title Stochastic Programming E-Print Series local.edoc.type-name Buch local.edoc.container-type series local.edoc.container-type-name Schriftenreihe local.edoc.container-publisher-name Springer local.edoc.container-publisher-place Berlin local.edoc.container-volume 2003 local.edoc.container-issue 3 local.edoc.container-year 2004 local.edoc.container-erstkatid 2936317-2