A cross-decomposition scheme with integrated primal-dual multi-cuts for two-stage stochastic programming investment planning problems

We describe a decomposition algorithm that combines Benders and scenariobased Lagrangean decomposition for two-stage stochastic programming investment planning problems with complete recourse, where the first-stage variables are mixedinteger and the second-stage variables are continuous. The algorithm is based on the 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 an illustrative case study for a facility location problem under disruptions, where the underlying LP relaxation is weak, and hence, multicut Benders decomposition converges only slowly. If the LP relaxation is improved by adding a tightening constraint, the performance of multi-cut Benders decomposition improves but the cross-decomposition scheme stays competitive and outperforms Benders for the largest problem instance.