High-dimensional Multi-objective Bayesian Optimization with Learned Variable Interactions

Multi-objective Bayesian optimization (MOBO) provides a principled framework for optimizing multiple expensive black-box functions. However, existing MOBO methods often struggle with coverage, scalability, and handling constraints and preferences. In this work we propose STAGE-BO, Sequential Targeting Adaptive Gap-Filling $\varepsilon$-Constraint Bayesian Optimization: by analyzing the coverage of the surrogate Pareto front, our method identifies the Pareto front point with the largest uncovered gap, and uses its coordinates to define adaptive constraints in $\varepsilon$-constraint method, which transforms the problem into a sequence of inequality-constrained subproblems, efficiently solved via constrained expected improvement acquisition. Our approach provides uniform Pareto coverage without hypervolume computation and naturally handles constraints and preferences. Experiments on synthetic and real-world benchmarks demonstrate superior coverage and competitive hypervolume performance against state-of-the-art baselines. Our code implementation can be found at https://github.com/YangYaohong1/STAGE-BO.

We consider grey-box optimization problems where the decision variables naturally partition into black-box variables (as arguments to an expensive black-box function) and white-box variables, governed by a set of explicit, closed-form equations that also depend on the output of the black-box function. We exploit this separability through a bilevel reformulation: an outer Bayesian optimization (BO) to optimize the scalar objective as a function of black-box variables alone, while an inner problem solves the white-box subproblem via global optimization. The Gaussian process surrogate used in BO is therefore defined rather than and white-box constraints are satisfied exactly whenever the inner optimizer converges to a feasible point---without penalty functions, chance constraints, or moment approximations. On a suite of 13 benchmark problems, bilevel BO achieves lower regret, with fewer iterations and wall clock time. This advantage is robust to initialization set size, exploration parameters, and inner-solver choice.

Designing effective multi-objective Bayesian optimization (MOBO) algorithms requires balancing many interdependent design choices whose optimal configuration is problem-dependent and typically demands deep expertise. We extend the LLaMEA framework to MOBO, using large language models as mutation and crossover operators within evolutionary strategies to generate complete algorithm implementations, with SMAC hyperparameter optimization integrated into the evolutionary loop. Across nine evolutionary runs we generated approximately 900 algorithms and benchmarked them on twelve synthetic problems (ZDT, DTLZ, WFG) and three real-world engineering problems (RE), using a BoFire qParEGO implementation as a state-of-the-art Bayesian-optimization baseline. On the synthetic suite the strongest generated algorithm attains the highest mean normalized hypervolume (0.971, vs. 0.869 for qParEGO) while requiring roughly 60x less wall-clock time; a Friedman test with post-hoc analysis places the two in a single top-performing group, and per-problem tests find the generated algorithm significantly better than qParEGO on 7 of the 12 problems and never worse, matching state-of-the-art accuracy at an order-of-magnitude lower cost. On the three unseen real-world engineering problems a generated algorithm attains the best mean normalized hypervolume (0.985, vs. 0.971 for qParEGO)--significantly better than qParEGO on two of the three problems--at roughly 3.4x lower wall-clock cost, confirming that the gains transfer beyond the synthetic regime. LLM-driven evolutionary search can thus discover algorithm designs that achieve Pareto-efficient trade-offs difficult to reach through manual design.