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Max-value Entropy Search for Multi-Objective Bayesian Optimization with Constraints
Syrine Belakaria, Aryan Deshwal, Janardhan Rao Doppa
TL;DR
The paper addresses constrained multi-objective black-box optimization when evaluations are expensive and the goal is to approximate feasible Pareto-optimal solutions efficiently. It proposes MESMOC, an output-space entropy acquisition method that selects evaluations for information about the optimal Pareto front. In two engineering applications, MESMOC outperforms existing baselines, including higher feasibility in aviation design and higher conversion efficiency in circuit design.
Problem
Constrained multi-objective optimization must approximate a feasible Pareto set with expensive evaluations, conflicting objectives, and black-box constraints.
Method
MESMOC uses an output-space entropy acquisition function to select candidate inputs that maximize information gain about the optimal Pareto front while satisfying constraints.
Results
MESMOC outperforms existing baselines in two engineering applications, with 50% feasible selections in aviation design and 88.81% conversion efficiency in circuit design.
Takeaways & Limitations
Output-space entropy search provides a practical approach for constrained multi-objective design across discrete and continuous engineering settings.
Takeaways & Limitations
The underlying information-gain formulation involves a high-dimensional distribution, expensive optimization, approximation error, and dependence on Monte Carlo sample count.
Abstract
from arXiv · showhide
We consider the problem of constrained multi-objective blackbox optimization using expensive function evaluations, where the goal is to approximate the true Pareto set of solutions satisfying a set of constraints while minimizing the number of function evaluations. For example, in aviation power system design applications, we need to find the designs that trade-off total energy and the mass while satisfying specific thresholds for motor temperature and voltage of cells. This optimization requires performing expensive computational simulations to evaluate designs. In this paper, we propose a new approach referred as {\em Max-value Entropy Search for Multi-objective Optimization with Constraints (MESMOC)} to solve this problem. MESMOC employs an output-space entropy based acquisition function to efficiently select the sequence of inputs for evaluation to uncover high-quality pareto-set solutions while satisfying constraints. We apply MESMOC to two real-world engineering design applications to demonstrate its effectiveness over state-of-the-art algorithms.
1 Introduction
Constrained multi-objective optimization combines expensive evaluations, conflicting objectives, and limited prior work on constrained Bayesian optimization. MESMOC addresses this setting with a cheaper output-space entropy acquisition function that targets information about the optimal Pareto front.
- Motivation: Engineering and scientific multi-objective problems require expensive experiments because objective functions are unknown.Examples include computational simulations for compiler optimization and physical laboratory experiments for material design.
- Motivation: Conflicting objectives prevent all objectives from being optimized simultaneously.
- Research gap: Constrained multi-objective Bayesian optimization has relatively limited prior work compared with single-objective Bayesian optimization.PESMOC is identified as the current state-of-the-art method for this problem setting.
- Research gap: PESMOC uses input-space entropy search, but its acquisition function is expensive to optimize and approximations may reduce accuracy.The method performs a series of approximations to improve efficiency, potentially at the expense of accuracy.
- Proposed approach: MESMOC selects evaluations by maximizing information gain about the optimal Pareto front while satisfying constraints.Its output-space entropy acquisition function is described as tighter to approximate and significantly cheaper to compute than input-space entropy approaches.
2 Background and Problem Setup
The paper formulates constrained multi-objective Bayesian optimization over unknown objectives and black-box constraints, with the goal of approximating the optimal Pareto set using few evaluations. Bayesian optimization combines a surrogate model, an acquisition function, and an optimization procedure to select candidates efficiently.
- Bayesian optimization: Bayesian optimization learns a cheap surrogate model from past evaluations and selects future inputs through exploration–exploitation trade-offs.
- Bayesian optimization: A Gaussian process models the unknown objective function using a mean function and covariance or kernel function.
- Bayesian optimization: An acquisition function scores candidate inputs according to the statistical model, using criteria such as EI, UCB, PES, or MES.
- Bayesian optimization: An optimization procedure selects the highest-scoring candidate input according to the acquisition function.DIRECT is given as a popular approach for optimizing the acquisition function.
- Problem formulation: The constrained multi-objective problem minimizes K ≥2 objectives subject to L black-box constraints and seeks the optimal Pareto set of feasible, nondominated points.The goal is to approximate this set while minimizing the number of function evaluations.
3 MESMOC for Multi-Objective Optimization with Constraints
MESMOC selects evaluations by maximizing output-space information gain about the constrained Pareto front. It combines Gaussian-process surrogates, sampled constrained multi-objective optimization, and entropy calculations to construct an efficient acquisition function.
- Computing Pareto front samples via cheap multi-objective optimization: For each entropy sample, MESMOC samples objective and constraint functions from their posterior models and solves a cheap constrained multi-objective problem to obtain a sample Pareto front.The implementation uses random Fourier features for sampled functions and constrained NSGA-II as the cheap solver, while allowing other solvers.
- Surrogate models: The algorithm models objectives and constraints with independent Gaussian processes trained on prior evaluations, whose outputs form the observed vector for each input.The training data contain evaluated inputs together with objective and constraint values, and the surrogate models are updated iteratively.
- Output space entropy based acquisition function: Input-space entropy methods require a high-dimensional posterior over the optimal Pareto set and can be expensive, approximation-dependent, and sensitive to Monte Carlo sample count.The paper identifies computational cost, potentially sub-optimal approximations, and sample sensitivity as key challenges.
- Output space entropy based acquisition function: MESMOC computes an output-space entropy acquisition function that selects candidate inputs by maximizing information gain about the constrained Pareto front.This extends output-space entropy search to constrained multi-objective optimization, avoiding direct inference over the optimal Pareto set.
- Entropy computation with a sample Pareto front: MESMOC approximates the expectation over sampled Pareto fronts with Monte Carlo and is designed for computational efficiency and robustness to the number of samples.The acquisition function is evaluated from sampled Pareto fronts before each expensive function evaluation, after which the models are updated.
- Entropy computation with a sample Pareto front: Given a sampled Pareto front, the method bounds each output component by its maximum front value and models that component with a truncated Gaussian distribution.Because the functions are modeled as independent Gaussian processes, the joint entropy decomposes into a sum of component entropies.
4 Experiments and Results
MESMOC is evaluated against PESMOC, MOEAD, and NSGA-II on electrified aviation power-system and analog-circuit design tasks. It achieves better Pareto-front quality and constraint-aware performance, including 88.81% circuit efficiency and higher feasible-design rates in the aviation task.
- Experimental setup: MESMOC is evaluated on electrified aviation power-system design and analog-circuit design optimization using GP models and comparisons with PESMOC, MOEAD, and NSGA-II.The experiments use squared exponential-kernel Gaussian processes, with hyperparameters estimated after every five function evaluations.
- Electrified aviation power system design: The aviation task minimizes mass and total energy under five constraints, with only 9% of 250,000 designs feasible and five points on the optimal Pareto front.The constraints include limits on discharge, cell voltage, motor temperature, inverter temperature, and modulation index.
- Analog circuit design optimization: The analog-circuit task uses 33 continuous input variables and 15 constraints, with results assessed using Pareto hypervolume and efficiency-prioritized circuit designs.Because satisfying every constraint is impossible in this setting, hypervolume includes all evaluated points.
- Overall results: MESMOC outperforms existing baselines on Pareto hypervolume, recovering a better Pareto front with a significant gain in the number of function evaluations.The hypervolume metric is plotted against the number of function evaluations in Figure 1.
- Electrified aviation power system design: 50% of MESMOC-selected aviation designs satisfy all constraints, compared with 1.5% for PESMOC, 9.5% for MOEAD, and 7.5% for NSGA-II.MESMOC did not recover all five optimal Pareto points but closely approximated the true Pareto front and recovered better designs than the baselines.
- Analog circuit design optimization: 88.81% conversion efficiency is achieved by the MESMOC-optimized circuit, improving on PESMOC by 12.61% and on NSGA-II by 17.86% with similar output ripples.The optimized circuit also meets the voltage-reference requirements and produces target outputs across the reported load ranges.