On the ReLU Lagrangian Cuts for Stochastic Mixed Integer Programming

TL;DR

This paper introduces ReLU Lagrangian cuts for stochastic mixed integer programs, enabling more efficient optimization by reformulating nonanticipativity constraints with ReLU functions and providing closed-form solutions.

Contribution

The paper presents a novel family of Lagrangian cuts using ReLU functions, which can be integrated into scenario decomposition methods for stochastic mixed integer programming.

Findings

ReLU Lagrangian cuts achieve optimality in stochastic mixed integer programs.

Closed-form expressions for the cuts are derived without solving dual problems.

Numerical results show the cuts outperform existing methods.

Abstract

We study stochastic mixed integer programs with both first-stage and recourse decisions involving mixed integer variables. 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 methods. We show that including ReLU Lagrangian cuts is sufficient to achieve optimality in the original stochastic mixed integer programs. 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 methods to enhance the cut coefficients. Numerical studies demonstrate the effectiveness of the proposed cuts compared to existing cut families.