A new interior-point approach for large two-stage stochastic problems

Abstract

Two-stage stochastic models give rise to very large optimization problems. Several approaches havebeen devised for efficiently solving them, including interior-point methods (IPMs). However, usingIPMs, the linking columns associated to first-stage decisions cause excessive fill-in for the solutionof the normal equations. This downside is usually alleviated if variable splitting is applied to first-stage variables. This work presents a specialized IPM that applies variable splitting and exploits thestructure of the deterministic equivalent of the stochastic problem. The specialized IPM combinesCholesky factorizations and preconditioned conjugate gradients for solving the normal equations.This specialized IPM outperforms other approaches when the number of first-stage variables is largeenough. This paper provides computational results for two stochastic problems: (1) a supply chainsystem and (2) capacity expansion in an electric system. Both linear and convex quadratic formu-lations were used, obtaining instances of up to 38 million variables and six million constraints. Thecomputational results show that our procedure is more efficient than alternative state-of-the-art IPMimplementations (e.g., CPLEX) and other specialized solvers for stochastic optimization