# On two-stage convex chance constrained problems

## 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.

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