Mathematical Methods of Operations Research
, Volume 65, Issue 1, pp 115140
First online:
On twostage convex chance constrained problems
 E. ErdoğanAffiliated withIEOR Department, Columbia University Email author
 , G. IyengarAffiliated withIEOR Department, Columbia University
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In this paper we develop approximation algorithms for twostage convex chance constrained problems. Nemirovski and Shapiro (Probab Randomized Methods Des Uncertain 2004) formulated this class of problems and proposed an ellipsoidlike iterative algorithm for the special case where the impact function f (x, h) is biaffine. We show that this algorithm extends to biconvex 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 twostage chance constrained problem—the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous twostage 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 twostage chance constrained problem when the impact function f (x, h) is biaffine and the extreme points of a certain “dual” polytope are known explicitly.
 Title
 On twostage convex chance constrained problems
 Journal

Mathematical Methods of Operations Research
Volume 65, Issue 1 , pp 115140
 Cover Date
 200702
 DOI
 10.1007/s0018600601042
 Print ISSN
 14322994
 Online ISSN
 14325217
 Publisher
 SpringerVerlag
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 Authors

 E. Erdoğan ^{(1)}
 G. Iyengar ^{(1)}
 Author Affiliations

 1. IEOR Department, Columbia University, NY, New York, 10027, USA