Abstract
We describe a cross-decomposition algorithm that combines Benders and scenario-based Lagrangean decomposition for two-stage stochastic programming investment planning problems with complete recourse, where the first-stage variables are mixed-integer and the second-stage variables are continuous. The algorithm is a novel cross-decomposition scheme and fully integrates primal and dual information in terms of primal---dual multi-cuts added to the Benders and the Lagrangean master problems for each scenario. The potential benefits of the cross-decomposition scheme are demonstrated with numerical experiments on a number of instances of a facility location problem under disruptions. In the original formulation, where the underlying LP relaxation is weak, the cross-decomposition method outperforms multi-cut Benders decomposition. If the formulation is improved with the addition of tightening constraints, the performance of both decomposition methods improves but cross-decomposition clearly remains the best method for large-scale problems.
Highlights
In the following we focus on decomposition schemes, which have been successfully applied for solving two-stage stochastic programming problems
We have described a cross-decomposition algorithm that combines Benders and scenariobased Lagrangean decomposition for two-stage stochastic MILP problems with complete recourse, where the first-stage variables are mixed-integer and the second-stage variables are continuous
The algorithm fully integrates primal and dual information with multi-cuts that are added to the Benders and the Lagrangean master problems for each scenario
Summary
Two-stage stochastic programming investment planning problems (Birge and Louveaux, 2011) can be hard to solve since the resulting deterministic equivalent programs can lead to very large-scale problems. On the other hand, Sohn et al (2011) derive a mean value cross-decomposition approach for two-stage stochastic programming problems (LP) based on Holmberg’s scheme (1992), in which the use of any master problem is eliminated and apply the algorithm to a set of random problem instances. They claim to solve the instances faster than Benders and ordinary cross-decomposition. The primal search is guided by a multi-cut Benders master problem, which is augmented with optimality cuts obtained from the Lagrangean dual subproblems.
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