Simulated Annealing-based Candidate Optimization for Batch Acquisition Functions

TL;DR

This paper addresses the challenge of optimizing batch acquisition functions in multi-objective Bayesian optimization when discrete candidate sets are required. It introduces a simulated annealing framework that maximizes the $q$-Expected Hypervolume Improvement ($q$EHVI) over a predefined candidate set, leveraging a Gaussian Process surrogate and both sequential and GPU-accelerated parallel implementations. Across four benchmark problems with varying dimensionality and objective counts, SA generally outperforms gradient-based methods (SLSQP) in final hypervolume and Pareto-front exploration, with particularly large gains on high-dimensional or nonlinear landscapes; Kursawe remains a case where gradient methods perform better. A real-world 9D materials-design campaign demonstrates the practical benefits of parallel SA, achieving faster convergence and higher final hypervolume than sequential SA. The work highlights the viability of metaheuristic approaches for robust, discrete-batch acquisition optimization in multi-objective Bayesian optimization and points to future enhancements such as adaptive cooling and distributed implementations.

Abstract

Bayesian Optimization with multi-objective acquisition functions such as q-Expected Hypervolume Improvement (qEHVI) requires efficient candidate optimization to maximize acquisition function values. Traditional approaches rely on continuous optimization methods like Sequential Least Squares Programming (SLSQP) for candidate selection. However, these gradient-based methods can become trapped in local optima, particularly in complex or high-dimensional objective landscapes. This paper presents a simulated annealing-based approach for candidate optimization in batch acquisition functions as an alternative to conventional continuous optimization methods. We evaluate our simulated annealing approach against SLSQP across four benchmark multi-objective optimization problems: ZDT1 (30D, 2 objectives), DTLZ2 (7D, 3 objectives), Kursawe (3D, 2 objectives), and Latent-Aware (4D, 2 objectives). Our results demonstrate that simulated annealing consistently achieves superior hypervolume performance compared to SLSQP in most test functions. The improvement is particularly pronounced for DTLZ2 and Latent-Aware problems, where simulated annealing reaches significantly higher hypervolume values and maintains better convergence characteristics. The histogram analysis of objective space coverage further reveals that simulated annealing explores more diverse and optimal regions of the Pareto front. These findings suggest that metaheuristic optimization approaches like simulated annealing can provide more robust and effective candidate optimization for multi-objective Bayesian optimization, offering a promising alternative to traditional gradient-based methods for batch acquisition function optimization.

Figures (5)

• Figure 1: Hypervolume convergence (left) and Pareto front analysis with marginal histograms (right) for ZDT1 ($D=30, O=2$). The simulated annealing approach (orange) demonstrates superior performance compared to SLSQP (blue), achieving higher final hypervolume values and continued improvement throughout the optimization process.
• Figure 2: Hypervolume convergence (left) and 3-D Pareto front projections with marginal histograms (right) for DTLZ2 ($D=7, O=3$). Simulated annealing shows dramatically superior performance, achieving significantly higher hypervolume values compared to SLSQP.
• Figure 3: Hypervolume convergence (left) and Pareto front analysis (right) for the Kursawe function ($D=3, O=2$). In this case, SLSQP demonstrates superior performance, achieving higher hypervolume values and faster convergence.
• Figure 4: Hypervolume convergence (left) and Pareto front scatter plus marginal histograms (right) for the latent-aware function ($D=4, O=2$). Simulated annealing demonstrates exceptional performance with dramatic improvements over SLSQP.
• Figure 5: Comparison of sequential versus parallel simulated annealing implementations for a real-world 5-objective materials optimization campaign. The parallel implementation (red) demonstrates superior convergence characteristics compared to the sequential approach (blue), achieving higher final hypervolume values and faster convergence through GPU-accelerated candidate evaluation.