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.

摘要

两阶段随机模型会产生非常大的优化问题。已经设计了几种方法来有效地解决它们，包括内点方法（IPM）。但是，使用 IPM，与第一阶段决策相关的链接列会导致对正规方程的解进行过多的填充。如果将变量拆分应用于第一阶段变量，则通常可以缓解这种不利影响。这项工作提出了一种专门的 IPM，它应用变量分裂并利用随机问题的确定性等价结构。专门的 IPM 结合了 Cholesky 分解和预条件共轭梯度来求解正规方程。当第一阶段变量的数量足够大时，这种专门的 IPM 优于其他方法。本文提供了两个随机问题的计算结果：（1）供应链系统和（2）电力系统的容量扩展。使用了线性和凸二次公式，获得了多达 3800 万个变量和 600 万个约束的实例。计算结果表明，我们的过程比替代的最先进的 IPM 实现（例如 CPLEX）和其他用于随机优化的专用求解器更有效。