Polynomial Cutting Plane Algorithms for Two-stage Stochastic Linear Programs Based on Ellipsoids, Volumetric Centers and Analytic Centers 1

Traditional simplex-basedalgorithms for two-stage stochastic linear programscan be broadly divided into two classes: (a) those that explicitly exploit the structure of the equivalent large-scale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly exploit that structure. Algorithms of class (b) are in general preferred. In 1988, following the work of Karmarkar for general linear programs, Birge and Qi 10] proposed a specialization of Karmarkar's algorithm for two-stage stochastic linear programs. The algorithm of Birge and Qi 10] is the rst interior point analog of class (a). Several other authors have studied related and diierent interior point analogs of class (a). Birge and Qi 10] also presented an analysis of the computational complexity of their algorithm. This analysis indicates that the computational complexity (in terms of total arithmetic operations) of their algorithm is in general smaller than that of the Karmarkar's algorithm (applied without modiication), and is quadratic in the number of realizations. Interior point analogs of the preferred class (b) have not received much attention. The only work on such algorithms is by Bahn, DU Merle, Goon and Vial 6], who present an algorithm based on analytic centers. However, they do not present results on the complexity of their algorithm. In this paper, we present three interior point analogs of class (b) based respectively on the ellipsoid algorithm (Khachiyan 26]), on the notion of volumetric center (Vaidya 35]) and on the notion of analytic center (Sonnevend 32]). We also present complexity results which indicate that the complexities (in terms of total arithmetic operations) of the three algorithms are linear in the number of realizations.