Multi-objective Bayesian optimization (MOBO) is commonly approached through specialized acquisition functions or scalarization schemes designed to explicitly account for trade-offs among non-preferential objectives. In this work, we show that such complexity might be unnecessary. We propose a framework that extends standard single-objective acquisition functions directly to the multi-objective setting through a hypervolume-based transformation. We further extend hedge strategies for acquisition functions, which are typically used only in single-objective optimization, to the multi-objective regime. Our approach requires minimal modification to existing Bayesian optimization pipelines and avoids the need for bespoke multi-objective formulations. We demonstrate how a broad class of commonly used single-objective acquisition functions and hedge strategies can be adapted in a principled manner to handle multiple objectives, while preserving their intuitive interpretation and computational efficiency. Empirically, we evaluate the proposed methods across a range of synthetic and real-world multi-objective benchmarks. Despite their simplicity, our extensions consistently match or outperform more complex state-of-the-art MOBO methods in terms of optimization performance and sample efficiency. These results suggest that effective multi-objective Bayesian optimization can be achieved by reusing and carefully extending well-established single-objective acquisition strategies, offering a simpler and more flexible alternative to existing approaches.
Figures & tables
Figure 1: Log-normalized hypervolume gap for batch optimization ( Q=3 ) across benchmarks. Lower values are better. Mean (solid line) ±1 standard deviation (shaded) over 20 runs.
Test Function
q HAX- UCB
q HAX- EI
q MOJES
q HVKG
q LNParEGO
q LNEHVI
ZDT2
230.41
226.79
1834.78
268.04
454.05
673.54
ZDT4
214.38
219.12
2817.48
296.25
506.43
680.57
ZDT6
221.89
214.73
8092.44
321.96
799.51
953.11
DTLZ1
254.31
246.49
4636.83
424.22
372.96
711.56
DTLZ2
243.24
243.02
4089.77
390.18
352.85
886.75
DTLZ3
220.59
218.35
2943.21
229.57
507.18
506.37
Table 1: Mean runtime in seconds for each algorithm and test function over 20 independent runs. (Batch mode: Q=3 )
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Test function
Dimensions ( dim )
No. of objectives ( M )
ZDT2
9
2
ZDT4
6
2
ZDT6
6
2
DTLZ1
8
3
DTLZ2
6
3
DTLZ3
4
2
Appendix
Table 2: Input dimensions and number of objectives for each benchmark test function.
Figure 2: Log-normalized inverted generational distance gap for batch optimization ( Q=3 ) across benchmarks. Lower values are better. Mean (solid line) ±1 standard deviation (shaded) over 20 runs.
Test Function
q HAX- UCB
q HAX- EI
q MOJES
q HVKG
q LNParEGO
q LNEHVI
ZDT2
676.62
715.49
2945.64
558.25
126.52
185.74
ZDT4
797.93
613.77
4469.76
668.50
265.70
186.82
ZDT6
597.61
612.06
17750.69
817.01
228.30
523.60
DTLZ1
685.13
717.91
6884.48
1074.03
378.68
453.49
DTLZ2
668.83
684.98
6523.09
959.74
270.95
574.83
DTLZ3
667.09
647.52
4010.11
615.54
167.90
127.60
Appendix
Table 3: Mean runtime in seconds for each algorithm and test function over 20 independent runs. (Sequential mode: Q=1 )
Figure 3: Log-normalized hypervolume gap for sequential optimization ( Q=1 ) across benchmarks. Lower values are better. Mean (solid line) ±1 standard deviation (shaded) over 20 runs.
Figure 4: Log-normalized IGD gap for sequential optimization ( Q=1 ) across benchmarks. Lower values are better. Mean (solid line) ±1 standard deviation (shaded) over 20 runs.
Multiobjective Bayesian optimization (MOBO) with Gaussian process (GP) surrogates is a sample efficient approach to solving multiobjective optimization problems. In MOBO, a Bayesian decision theoretic acquisition function guides the adaptive selection of new candidate inputs, on which objectives and constraints are evaluated to update the surrogate model sequentially. Existing approaches maintain independent GP models for the objectives and constraints, with new observations evaluating all objectives and constraints in a coupled fashion. However, the objectives and constraints often contain inherent correlations which, if exploited, can enable decoupled evaluations where only a subset of them are evaluated at each round. We present a new approach that leverages a multitask GP model to jointly learn all objectives and constraints, and propose a total correlation metric that enables identifying an optimal subset of objectives and constraints to be evaluated at every round, even under uniform evaluation costs. Theoretically, we show that our acquisition policy is asymptotically consistent despite decoupling and that our proposed decoupled subset selection rule maximizes the expected posterior entropy reduction about unevaluated tasks under mild conditions. Empirically, we show that our approach outperforms coupled and decoupled baselines in the state of the art.
Ashwin Renganathan, Peter Bachman
Aerospace Engineering Pennsylvania State University University Park, PA 16802 · Institute of Computational and Data Sciences (ICDS) Pennsylvania State University University Park, PA 16802
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 ε-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 ε-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.
Yaohong Yang, Sammie Katt, Samuel Kaski
Department of Computer Science, Aalto University, Espoo, Finland · ELLIS Institute Finland · Department of Computer Science, University of Manchester, Manchester, United Kingdom
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.