Volumetric Center Method for Stochastic Convex Programs using Sampling Sanjay Mehrotra 1 IE/MS Technical Report, December 1999 (minor revision January 2000) Department of Industrial Engineering and Management Sciences, Robert R. McCormick School of Engineering, Northwestern University, Evanston, Illinois 60208 [email protected]

Abstract We develop an algorithm for solving the stochastic convex program (SCP) by combining Vaidya's volumetric center interior point method (VCM) for solving non-smooth convex programming problems with the Monte-Carlo sampling technique to compute a subgradient. A near-central cut variant of VCM is developed, and for this method an approach to perform bulk cut translation, and adding multiple cuts is given. We show that by using near-central VCM the SCP can be solved to a desirable accuracy with any given probability. For the two-stage SCP the solution time is independent of the number of scenarios. Stochastic Programming, Volumetric Center, Analytic Center, Interior Point Methods, Convex Programming, Volumetric Center Key Words:

The author thanks the Supreme Lord for all revelations that lead to this paper. All its fruits are surrendered to Him. 1

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1 Introduction In this paper we develop an algorithm for solving the general stochastic convex problem 65, 12] (SCP): min c0(x) E r0(x
~)] s:t: ci(x) E ri(x
~)]
0 i = 1 : : : m x 2 X