Threshold Boolean Form for Joint Probabilistic Constraints with Random Technology Matrix

Alexander Kogan; Miguel A. Lejeune

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SPEPS Preprint

Author(s):	Alexander Kogan, Rutgers University Miguel A. Lejeune, George Washington University
Title:	Threshold Boolean Form for Joint Probabilistic Constraints with Random Technology Matrix
Date of Acceptance:	23.11.2012
Submission Date:	08.09.2012
Series Title:	Stochastic Programming E-Print Series (SPEPS)
Editors:	Julie L. Higle; Werner Römisch; Surrajeet Sen
Complete Preprint:	pdf (urn:nbn:de:kobv:11-100206028)
Keywords (eng):	Boolean function, stochastic programming, joint probabilistic constraint, random technology matrix, threshold function
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Abstract (eng):

We develop a new modeling and exact solution method for stochastic programming problems that include a joint probabilistic constraint in which the multi-row random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations equivalent to the studied stochastic problem. We derive a set of strengthening valid inequalities for the reformulated problems. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the MIP reformulations are much easier and orders of magnitude faster to solve than the MINLP formulation. The method integrating the derived valid inequalities in a branch-and-bound algorithm has the best performance.

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