Joint Surrogate Learning of Objectives, Constraints, and Sensitivities for Efficient Multi-objective Optimization of Neural Dynamical Systems

Discrete black-box optimization is often addressed using approaches such as Sequential Model-Based Optimization (SMBO), which aims to improve sample efficiency by fitting surrogate models that approximate a costly objective function over a discrete search space. In many real-world problems, the set of feasible inputs is often given by logical constraints known in advance. However, existing surrogate modeling techniques generally fail to capture the symbolic rules governing feasibility in discrete input spaces. In this paper, we propose a surrogate modeling approach based on tensor decomposition that captures the structure of discrete search spaces while directly integrating feasibility information. To implement this approach, we formulate surrogate model training as a constrained polynomial optimization problem and solve a relaxed formulation using a differentiable penalty term derived from T-norms. Our experiments on both synthetic and real-world benchmarks, including a pressure vessel design task, demonstrate that the proposed method improves sample efficiency by effectively guiding the search away from infeasible regions.

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.

The problem of multiobjective optimization under uncertainties is often approached by taking the expectation of each objective. In this work, we propose instead to formulate this as a Bayesian decision problem and to rely on the expected value of the hypervolume, which is to be maximized with respect to a finite set of input points. We show that this can be performed using methods based on gradients in a stochastic optimization framework, provided that care is taken with respect to dominated points. Moreover, in the absence of readily available differentiable code, we propose to use Gaussian Processes as differentiable surrogate models, in order to perform the optimization. An additional contribution in this work are some active learning strategies, through acquisition functions which helps construct a surrogate model well-designed for the multiobjective optimization problem at stake. These strategies are compared on simple analytical problems to assess their performances.