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

Two-stage stochastic models give rise to very large optimization problems. Several approaches have been devised for efficiently solving them, including interior-point methods (IPMs). However, using IPMs, the linking columns associated to first-stage decisions cause excessive fill-in for the solution of 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 the structure of the deterministic equivalent of the stochastic problem. The specialized IPM combines Cholesky factorizations and preconditioned conjugate gradients for solving the normal equations. This specialized IPM outperforms other approaches when the number of first-stage variables is large enough. This paper provides computational results for two stochastic problems: (1) a supply chain system and (2) capacity expansion in an electric system. Both linear and convex quadratic formulations were used, obtaining instances of up to 38 million variables and six million constraints. The computational results show that our procedure is more efficient than alternative state-of-the-art IPM implementations (e.g., CPLEX) and other specialized solvers for stochastic optimization.

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Research Report UPC-DEIO-JC DR 2020-01, Dept. of Statistics and Operations Research, Universitat Polit├Ęcnica de Catalunya, Barcelona, Catalonia, April 2020.

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