A new interior-point approach for large separable convex quadratic two-stage stochastic problems. (4th May 2022)

• Record Type: Journal Article
• Authors: Castro, Jordi; de la Lama-Zubirán, Paula
• Is Part Of: Optimization methods and software. Volume 37:Number 3(2022)
• Page Start: 801
• Page End: 829
• Publication Date: 2022-05-04
• Subjects: Interior-point methods -- stochastic optimization -- structured problems -- large-scale optimization; 90C51 -- 90C15 -- 90C06; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2020.1841190
• Languages: English
• ISSNs: 1055-6788

Abstract

ABSTRACT: 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 with 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 6 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.