[Paper Review] Budgeted Multi-Objective Optimization with a Focus on the Central Part of the Pareto Front -- Extended Version

David Gaudrie, Rodolphe Le Riche+3 more|arXiv (Cornell University)|Sep 27, 2018

Advanced Multi-Objective Optimization Algorithms83 references4 citations

TL;DR

This paper proposes C-EHI, a Bayesian multi-objective optimization algorithm that focuses on the central region of the Pareto front to improve solution quality and coverage under strict budget constraints. By dynamically targeting well-balanced solutions using a center-based Expected Hypervolume Improvement criterion and adapting the search region based on remaining budget, C-EHI achieves superior convergence to the central Pareto front compared to state-of-the-art methods on benchmark and real-world aerodynamic problems.

ABSTRACT

Optimizing nonlinear systems involving expensive computer experiments with regard to conflicting objectives is a common challenge. When the number of experiments is severely restricted and/or when the number of objectives increases, uncovering the whole set of Pareto optimal solutions is out of reach, even for surrogate-based approaches: the proposed solutions are sub-optimal or do not cover the front well. As non-compromising optimal solutions have usually little point in applications, this work restricts the search to solutions that are close to the Pareto front center. The article starts by characterizing this center, which is defined for any type of front. Next, a Bayesian multi-objective optimization method for directing the search towards it is proposed. Targeting a subset of the Pareto front allows an improved optimality of the solutions and a better coverage of this zone, which is our main concern. A criterion for detecting convergence to the center is described. If the criterion is triggered, a widened central part of the Pareto front is targeted such that sufficiently accurate convergence to it is forecasted within the remaining budget. Numerical experiments show how the resulting algorithm, C-EHI, better locates the central part of the Pareto front when compared to state-of-the-art Bayesian algorithms.

Motivation & Objective

• To address the challenge of optimizing expensive, multi-objective systems with limited evaluations, where full Pareto front approximation is infeasible.
• To improve solution quality and coverage in the central, well-balanced region of the Pareto front, which is often most relevant for decision-makers.
• To develop a preference-free, automatic method for focusing optimization on the central part of the Pareto front using surrogate models.
• To introduce a dynamic mechanism that widens the target region when convergence to the center is detected, maximizing utility of remaining budget.

Proposed method

• The Pareto front center is defined as the point on the estimated Ideal-Nadir line closest to the empirical Pareto front, computed from Gaussian process surrogates.
• A modified Expected Hypervolume Improvement (EHI) criterion is used to prioritize sampling points that improve the central region of the front.
• A convergence detection criterion is introduced to identify when the central region is sufficiently well-approximated, triggering a shift in focus.
• When convergence is detected, the search region is widened to target a broader central segment, ensuring remaining budget is used effectively.
• The method integrates with existing Bayesian optimization frameworks and uses probabilistic dominance and extreme point estimation to guide selection.
• The algorithm dynamically updates the target region based on both the current Pareto front estimate and remaining computational budget.

Experimental results

Research questions

• RQ1 Can focusing optimization on the central region of the Pareto front yield better solutions under strict budget constraints than full-front approaches?
• RQ2 How can the central region of the Pareto front be automatically defined and tracked using only surrogate model estimates?
• RQ3 What criterion enables reliable detection of convergence to the central Pareto front during optimization?
• RQ4 How should the search region be adapted when convergence to the center is achieved, to maximize remaining budget efficiency?
• RQ5 How does the proposed C-EHI method compare in performance to state-of-the-art Bayesian multi-objective optimization algorithms on real-world and benchmark problems?

Key findings

• C-EHI achieves significantly better convergence to the central Pareto front than state-of-the-art Bayesian algorithms on the ZDT1 and P1 benchmark problems.
• On real-world airfoil aerodynamic problems with 3, 8, and 22 design variables and 2–4 objectives, C-EHI demonstrates improved coverage and optimality in the central region.
• The proposed convergence detection criterion reliably identifies when the central region is well-approximated, enabling timely expansion of the target region.
• Widening the search region after convergence detection leads to more accurate and comprehensive coverage of the central front within the remaining budget.
• The method outperforms standard EHI and other preference-based approaches in locating well-balanced, compromise solutions without requiring user-specified preferences.
• The algorithm maintains strong performance across diverse problem dimensions and objective counts, including high-dimensional cases with up to 22 variables.