ILP-BO: Integer Linear Programming-Based Black-Box Optimization

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.

Integer Linear Programming (ILP) serves as a versatile framework for modeling a wide range of combinatorial optimization problems, typically addressed by sophisticated exact solvers or heuristics. While learning-based approaches have recently shown their effectiveness, they suffer from poor generalization to out-of-distribution instances and inherent dependence on external solvers. In this work, we propose a solver-free, sampling-based optimization framework for ILP that directly explores discrete feasible regions without training or external solvers. Exploiting the linear structure of ILP, we employ a Locally-Balanced Proposal to construct a transition kernel, thereby avoiding the gradient approximation. To overcome the highly multimodal nature of ILP energy landscapes, we integrate Parallel Tempering. In addition to standard temperature tempering, we introduce penalty tempering, which modulates constraint barriers while preserving the objective landscape over feasible solutions. Empirically, our method consistently outperforms SCIP across all four benchmarks, matches or exceeds Gurobi on two of four tasks within a 200-second budget, and is substantially more robust to distribution shift than learning-based methods. Furthermore, on MIPLIB 2017 instances, our framework remains competitive with classical solvers without any problem-specific tuning.

Many real-world tasks require optimizing expensive black-box functions accessible only through noisy evaluations, a setting commonly addressed with Bayesian optimization. While Gaussian processes(GPs) and Bayesian neural networks have driven significant progress, GP-based methods scale cubically with observations, and BNN-based approaches remain burdened by expensive posterior sampling such as Hamiltonian Monte Carlo, and inner-loop acquisition optimization in high-dimensional setting. In this work, we propose VBO-MI, a fully gradient-based BO framework that requires no assumptions on a Gaussian prior or a variational posterior family over $f$. While our framework uses gradient-based updates to train the internal action and information-estimation networks, it treats the objective function $f$ as a strict black box, requiring only zeroth-order evaluations. To enable end-to-end gradient flow, we employ an actor-critic architecture consisting of an action-net to navigate the input space and a variational critic to estimate a surrogate information signal for exploration. Additionally, we design a lightweight surrogate that approximates the objective from observed data using a replay buffer, which reduces the total required real function queries to one batch per iteration. We provide theoretical analysis for our framework, establishing convergence of its training objective and surrogate estimator, and showing, that near-optimal solutions exist. We evaluate our method on a diverse suite of benchmarks, including high-dimensional synthetic function and complex real-world tasks such as Rover Trajectory planning optimization, the Lunar Lander, and Pest Control problems. Our experiments demonstrate that VBO-MI provides the same or superior optimization performance over the various baselines.