# Improved Max-value Entropy Search for Multi-objective Bayesian Optimization with Constraints

arXiv: https://arxiv.org/abs/2011.01150 · HTML: https://mlsanity.com/paper/2011.01150/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2020.

## Built on by (most cited first, 15 of 15)

- [Unexpected Improvements to Expected Improvement for Bayesian Optimization](https://mlsanity.com/paper/2310.20708/) (2023)
- [GIBBON: General-purpose Information-Based Bayesian OptimisatioN](https://mlsanity.com/paper/2102.03324/) (2021)
- [Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound](https://mlsanity.com/paper/2102.09788/) (2021)
- [Bayesian Optimization over Hybrid Spaces](https://mlsanity.com/paper/2106.04682/) (2021)
- [Output Space Entropy Search Framework for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2110.06980/) (2021)
- [OVD-Explorer: Optimism Should Not Be the Sole Pursuit of Exploration in Noisy Environments](https://mlsanity.com/paper/2312.12145/) (2023)
- [Constrained Multi-objective Bayesian Optimization through Optimistic Constraints Estimation](https://mlsanity.com/paper/2411.03641/) (2024)
- [PF2ES: Parallel Feasible Pareto Frontier Entropy Search for Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2204.05411/) (2022)
- [Preference-Aware Constrained Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2303.13034/) (2023)
- [Information-theoretic Bayesian Optimization: Survey and Tutorial](https://mlsanity.com/paper/2502.06789/) (2025)
- [Bayesian multi-objective optimization for stochastic simulators: an extension of the Pareto Active Learning method](https://mlsanity.com/paper/2207.03842/) (2022)
- [Preferential Multi-Objective Bayesian Optimization for Drug Discovery](https://mlsanity.com/paper/2503.16841/) (2025)
- [Local Entropy Search over Descent Sequences for Bayesian Optimization](https://mlsanity.com/paper/2511.19241/) (2025)
- [Non-Myopic Multi-Objective Bayesian Optimization](https://mlsanity.com/paper/2412.08085/) (2024)
- [High-dimensional Multi-objective Bayesian Optimization with Learned Variable Interactions](https://mlsanity.com/paper/2608.11713/) (2026)

## Abstract (arXiv)

We present MESMOC+, an improved version of Max-value Entropy search for Multi-Objective Bayesian optimization with Constraints (MESMOC). MESMOC+ can be used to solve constrained multi-objective problems when the objectives and the constraints are expensive to evaluate. MESMOC+ works by minimizing the entropy of the solution of the optimization problem in function space, i.e., the Pareto frontier, to guide the search for the optimum. The cost of MESMOC+ is linear in the number of objectives and constraints. Furthermore, it is often significantly smaller than the cost of alternative methods based on minimizing the entropy of the Pareto set. The reason for this is that it is easier to approximate the required computations in MESMOC+. Moreover, MESMOC+'s acquisition function is expressed as the sum of one acquisition per each black-box (objective or constraint). Thus, it can be used in a decoupled evaluation setting in which one chooses not only the next input location to evaluate, but also which black-box to evaluate there. We compare MESMOC+ with related methods in synthetic and real optimization problems. These experiments show that the entropy estimation provided by MESMOC+ is more accurate than that of previous methods. This leads to better optimization results. MESMOC+ is also competitive with other information-based methods for constrained multi-objective Bayesian optimization, but it is significantly faster.