Integer L-Shaped Method with Non-Supporting No-Good Optimality Cuts

Abstract: Two-stage stochastic mixed-integer linear programs with mixed-integer recourse arise in many practical applications but are computationally challenging due to their large size and the presence of integer decisions in both stages. The integer L-shaped method with alternating cuts is a widely used decomposition algorithm for these problems, relying on optimality cuts generated using subproblems to iteratively refine the master problem. A key computational bottleneck in this approach is solving the mixed-integer subproblems to optimality in order to generate separating cuts. This work proposes a modification to the integer L-shaped method with alternating cuts to allow for efficient generation of no-good optimality cuts that are separating for the current master problem solutions without being supporting hyperplanes of the feasible region. These separating cuts are derived from subproblems that are terminated before the optimal solution is found or proven to be optimal, reducing the computational effort required for cut generation. Additionally, an updated optimality cut generation function is proposed to account for the various complexities introduced by this early termination strategy. The effectiveness of the proposed method is demonstrated through two case studies on industrially relevant problems from the literature, which illustrate its advantages in handling large-scale instances with complex mixed-integer subproblems. In these cases, the method achieves substantial reductions in solution time or optimality gap compared to the standard integer L-shaped method with alternating cuts, with performance improvements that increase with mixed-integer subproblem size and complexity.