CORC Technical Report TR-2005-2 On two-stage convex chance constrained problems E.Erdo˘gan ∗ G. Iyengar † June 12, 2005 Abstract In this paper we develop approximation algorithms for two-stage convex chance constrained problems. Nemirovski and Shapiro [16] 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 [16]. 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. 1 Introduction The simplest model for a convex chance constrained problem is as follows. min c T x s.t. x ∈X ǫ (Q)=  y ∈X| Q(H : f (y, H) > 0) ≤ ǫ  , (1) * IEOR Department, Columbia University, New York, New York 10027. Email: ee168@columbia.edu. Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514. † IEOR Department, Columbia University, New York, New York 10027. Email: gi10@columbia.edu. Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514. 1