On two-stage convex chance constrained problems Original Article

First Online: 23 September 2006 Received: 03 November 2005 Accepted: 05 April 2006 DOI :
10.1007/s00186-006-0104-2

Cite this article as: Erdoğan, E. & Iyengar, G. Math Meth Oper Res (2007) 65: 115. doi:10.1007/s00186-006-0104-2 Abstract In this paper we develop approximation algorithms for two-stage convex chance constrained problems. Nemirovski and Shapiro (Probab Randomized Methods Des Uncertain 2004) formulated this class of problems and proposed an ellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine. We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm (Nemirovski and Shapiro in Probab Randomized Methods Des Uncertain 2004). Since the polytope determining r is random, computing r is difficult. Yet, the solution algorithm requires r as an input. In this paper we provide some guidance for selecting r . We show that the largest value of r is determined by the degree of robust feasibility of the two-stage chance constrained problem—the more robust the problem, the higher one can set the parameter r . Next, we formulate ambiguous two-stage chance constrained problems. In this formulation, the random variables defining the chance constraint are known to have a fixed distribution; however, the decision maker is only able to estimate this distribution to within some error. We construct an algorithm that solves the ambiguous two-stage chance constrained problem when the impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope are known explicitly.

Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.

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Authors and Affiliations 1. IEOR Department Columbia University New York USA