4. Optimization Models with Probabilistic Constraints

Abstract

Previous chapter Next chapter MOS-SIAM Series on Optimization Lectures on Stochastic Programming4. Optimization Models with Probabilistic ConstraintsDarinka DentchevaDarinka Dentchevapp.87 - 153Chapter DOI:https://doi.org/10.1137/1.9780898718751.ch4PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt 4.1 Introduction In this chapter, we discuss stochastic optimization problems with probabilistic (also called chance) constraints of the form Min c (x) s.t. Pr { gj (x,Z)≤0,j ∈ J }≥p, x ∈ X. 4.1 Here X⊂ ℝn is a nonempty set, c : ℝn →ℝ, gj : ℝn × ℝs →ℝ,j ∈ J, where J is an index set, Z is an s-dimensional random vector, and p is a modeling parameter. We denote by Pz the probability measure (probability distribution) induced by the random vector Z on ℝs. The event A (x)= { gj (x,Z)≤0,j ∈ J } in (4.1) depends on the decision vector x, and its probability Pr{A(x)} is calculated with respect to the probability distribution Pz. This model reflects the point of view that for a given decision x we do not reject the statistical hypothesis that the constraints gj (x,Z)≤0,j ∈ J , are satisfied. We discussed examples and a motivation for such problems in Chapter 1 in the contexts of inventory, multiproduct, and portfolio selection models. We emphasize that imposing constraints on probability of events is particularly appropriate whenever high uncertainty is involved and reliability is a central issue. In such cases, constraints on the expected value may not be sufficient to reflect our attitude to undesirable outcomes. We also note that the objective function c(x) can represent an expected value function, i.e., c (x)=E [ƒ (x,Z) ] ; however, we focus on the analysis of the probabilistic constraints at the moment. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-687-0eISBN:978-0-89871-875-1 https://doi.org/10.1137/1.9780898718751Book Series Name:MOS-SIAM Series on OptimizationBook Code:MP09Book Pages:xv + 431Key words:mathematical programming, stochastic optimization, convex analysis, risk analysis, modeling uncertainty