We study stochastic mixed integer programs where both first-stage and recourse decisions can be mixed integers. A new family of Lagrangian cuts, termed ``ReLU Lagrangian cuts," is introduced by reformulating the nonanticipativity constraints using ReLU functions. These cuts can be integrated into scenario decomposition algorithms. Unlike the ordinary Lagrangian cuts, we prove that the inclusion of ReLU Lagrangian cuts is sufficient to solve the original stochastic mixed integer programs to optimality. Without solving the Lagrangian dual problems, we derive closed-form expressions for these cuts. Furthermore, to speed up the cut-generating procedures, we introduce linear programming-based techniques to enhance the cut coefficients. Numerical studies demonstrate the effectiveness of the proposed cuts compared to existing methods.
Citation
Deng, H., Xie, W. (2024). On the ReLU Lagrangian Cuts for Stochastic Mixed Integer Programming. Available at Optimization Online.
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