In this paper, we explore the utilization of disjunctive cuts to enhance the strength of cuts within a Benders decomposition algorithm. At a broad level, the proposed algorithm leverages integrality information and Benders cuts to generate cuts tightly aligned with the convex hull of the extended formulation. Importantly, this is achieved via iteration of standard Benders subproblem. Specifically, our study highlights that normalization plays a crucial role in eliminating the generation of overly weak cuts. We conduct a comparative analysis to assess the impact of Benders disjunctive cuts under various normalization settings in capacitated facility location problems.

Multistage stochastic mixed-integer linear programming (MS-MILP) is a powerful framework for sequential decision-making under uncertainty. Solving MS-MILP with binary state variables traditionally involves iteratively solving MIP subproblems to generate Lagrangian cuts for approximating future value functions. However, this method can be highly time-consuming and computationally demanding, especially with complex subproblems. This study introduces an innovative LP-based Stochastic Dual Dynamic Programming (SDDP)-type algorithm specifically for MS-MILP problems with binary state variables. This approach eliminates the need to solve MIP subproblems. Using a geometric perspective of the connection between the convex envelope of a value function and the convex hull of its feasible region, which uses the cutting-planes tree algorithm to refine the LP relaxation of lifted subproblems' feasible region and generate Benders' cuts to approximate the value functions, leading to a more accurate approximation of the value function. A key achievement of this research is demonstrating the finite convergence of this algorithm, making it a more practical and efficient solution for complex MS-MILP problems. The complexity of solving a particular case of the problem, where uncertainty in the subproblems is eliminated from the constraints through reformulation, has been significantly reduced. This modification enables the efficient resolution of problems with extended time horizons and numerous realizations at each stage.

We present a cutting-plane-based decomposition algorithm along with several algorithmic enhancements designed to address two-stage distributionally robust optimization problems over 1-Wasserstein balls. Through a numerical analysis on a facility location problem with demand uncertainty, we highlight the potential benefits of employing a particular distance metric for enhanced out-of-sample performance, while also examining its computational trade-offs. Furthermore, we explore alternative approaches aimed at enhancing the scalability of our proposed algorithm.