DCEmbed: Scalable Optimization over Neural Surrogates

Abstract

Neural surrogates can accelerate large-scale optimization by replacing expensive or intractable model components with efficient learned approximations, but solving the resulting embedded problems can remain prohibitively costly. For instance, standard exact encodings of neural networks with ReLU activations allow the problem to be solved by mixed-integer solvers, but add large numbers of binary variables to accommodate the nonlinearity of the activations, which can render the problem computationally prohibitive. To address this, we propose DCEmbed, a heuristic for optimization problems with embedded neural surrogates that leverages the difference-of-convex (DC) representation of the network and avoids adding activation binaries. Exploiting shared structure within the DC representation of a ReLU network, we derive a reduced-size, exact formulation for its convex components that can be embedded in optimization problems using just two linear inequalities and one continuous auxiliary variable per hidden neuron. Using this, the problem is solved via an iterative penalty convex-concave procedure, where only the concave portions of the neural terms are approximated at each stage. The original objective, constraints, and any discrete decisions are retained, allowing standard convex or mixed-integer optimization solvers to optimize the host and surrogate jointly at each iteration. In experiments on quadratic programs, mixed-integer resource allocation, and neural two-stage stochastic programming, our method demonstrates much faster progress toward high-quality feasible solutions than approaches using exact mixed-integer embeddings. In particular, DCEmbed achieves $4\times$ lower normalized primal integral than the best exact baseline on the resource allocation problem, while in two-stage stochastic programming it reaches the global surrogate optimum $\sim 5\times$ faster than Gurobi ML.