Ambiguous Chance Constrained Programs: Algorithms and Applications

Ambiguous Chance Constrained Programs: Algorithms and Applications Emre Erdoğan Chance constrained problems are optimization problems where one or more constraints ensure that the probability of one or more events occurring is less than a prescribed threshold. Although it is typically assumed that the distribution defining the chance constraints are known perfectly; in practice this assumption is unwarranted. We study chance constrained problems where the underlying distributions are not completely specified and are assumed to belong to an uncertainty set Q. We call such problems “ambiguous chance constrained problems.” We focus primarily on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : ρp(Q,Q0) ≤ β}, where ρp denotes the Prohorov metric. We study single and two stage ambiguous chance constrained programs. The single stage ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q0. We show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem. We also show that the robust sampled problem can be solved efficiently both in theory and in practice. Nemirovski and Shapiro [61] formulated two-stage convex chance constrained programs 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 [61]. In this dissertation we provide some guidance for selecting r. We develop an approximation algorithm to two-stage ambiguous chance constrained programs when the impact function f(x,h) is bi-affine and the extreme points of a certain “dual” polytope are known explicitly.