Optimization over Trained Neural Networks: Difference-of-Convex Algorithm and Application to Data Center Scheduling

When solving decision-making problems with mathematical optimization, some constraints or objectives may lack analytic expressions but can be approximated from data. When an approximation is made by neural networks, the underlying problem becomes optimization over trained neural networks. Despite recent improvements with cutting planes, relaxations, and heuristics, the problem remains difficult to solve in practice. We propose a new solution based on a bilinear problem reformulation that penalizes ReLU constraints in the objective function. This reformulation makes the problem amenable to efficient difference-of-convex algorithms (DCA), for which we propose a principled approach to penalty selection that facilitates convergence to stationary points of the original problem. We apply the DCA to the problem of the least-cost allocation of data center electricity demand in a power grid, reporting significant savings in congested cases.