select via hypervolume

This work addresses a critical gap in preference-guided expected improvement criteria for multi-objective Bayesian optimization: the lack of a systematic understanding of the interplay among geometric structure, monotonicity, and exact computation. By analytically examining the geometric nature of hypervolume and the R2 indicator in both objective and scalarized spaces, the study reveals that the R2-based expected improvement corresponds to a volume in scalarized space rather than a weighted hypervolume in objective space. Building on this insight, the authors unify various hypervolume variants under a common computational perspective and introduce two exact calculation approaches—an ER2I algorithm based on finite summation for discrete settings and an integral-based method leveraging Gaussian surrogate models. Furthermore, they establish an achievement scalarizing optimization framework grounded in scalar Gaussian expected improvement and rigorously analyze its Pareto compliance and monotonicity properties.

In multi-objective Bayesian optimization, the high computational cost of hypervolume (HV) computation—especially in high-dimensional objective spaces—severely hampers acquisition function optimization. To address this, we propose an efficient and scalable box-decomposition-based approximation algorithm. Our method systematically constructs low-complexity approximations of HV increments, reducing worst-case memory complexity from superpolynomial to polynomial. This work provides, for the first time, a rigorous mathematical formulation of the approximation, a formal convergence analysis, and fully reproducible implementation details—thereby bridging critical gaps in theoretical rigor and algorithmic clarity present in prior literature. Empirical results demonstrate that our approach maintains solution-set quality while significantly accelerating HV improvement computation, particularly in high-dimensional objective spaces. The method thus offers practical scalability for large-scale multi-objective optimization.

This work addresses a critical gap in multi-objective optimization: the absence of a finite-set quality indicator that simultaneously guarantees strict Pareto compliance and effectively evaluates boundary points. The study introduces, for the first time, the concept of “magnitude” from metric geometry to construct a novel scalar indicator that is strictly Pareto compliant. By integrating coordinate projection with magnitude theory from category theory, the proposed metric exhibits both weak and strict set monotonicity and positively identifies boundary solutions—addressing key limitations of the hypervolume indicator. The associated algorithm achieves Θ(n log n) time complexity in two and three dimensions. Empirical results demonstrate that magnitude favors populations containing boundary points and complete Das–Dennis reference grids, whereas hypervolume tends to prefer densely filled interior configurations.

The R2 indicator for bi-objective optimization lacks strict Pareto compatibility—i.e., adding a dominated solution may not increase the indicator value. Method: This paper proposes an analytical variant of R2 based on a continuous uniform Tchebycheff utility function. We theoretically prove that this continuous R2 exhibits strict Pareto compatibility for bi-objective problems: adding any non-dominated solution strictly increases the indicator, and adding any dominated solution necessarily increases it. We further devise an exact O(N log N) algorithm, enabling the first efficient and theoretically compliant unary quality assessment. Results: Experiments show that the proposed indicator achieves evaluation performance comparable to hypervolume (HV), yet with significantly higher computational efficiency. It thus fills a critical gap in bi-objective set-quality indicators by simultaneously offering rigorous theoretical guarantees and practical scalability.

In multi-objective optimization, the Pareto-optimal solution set is often excessively large, imposing heavy cognitive burden on decision-makers. Method: This paper proposes “Directional Coverage,” a novel representativeness metric, and conducts axiomatic analysis within a multi-winner voting framework to expose counterintuitive behaviors of existing indicators and characterize their computational complexity boundaries under varying objective structures. The approach integrates axiomatic modeling, computational complexity theory, and empirical evaluation to systematically compare how diverse quality metrics affect solution set representativeness. Results: Experiments demonstrate that Directional Coverage achieves superior or comparable performance to state-of-the-art metrics in diversity, convergence, and directional sensitivity. Crucially, the choice of quality metric fundamentally determines representativeness outcomes—providing both theoretical foundations and practical tools for Pareto set reduction.

This work proposes a novel representation learning framework that addresses the limited representational capacity of existing methods in complex scenarios by integrating adaptive multi-scale fusion with contrastive learning. The approach dynamically aggregates multi-level semantic information and introduces a structure-aware contrastive loss, thereby significantly enhancing the model’s ability to discriminate fine-grained differences. Extensive experiments demonstrate that the proposed framework consistently outperforms state-of-the-art methods across multiple benchmark datasets, exhibiting particularly strong robustness under low-resource settings and in the presence of noise. Beyond advancing the theoretical foundations of representation learning, this study also delivers an efficient and scalable solution with practical applicability.

This work addresses the challenges of convergence in multi-objective optimization arising from non-differentiable objective functions and abrupt structural changes in nondominated fronts. To this end, the authors propose a nonsmooth set-based gradient ascent method that uniquely integrates an amplitude indicator with a hierarchical weighting scheme. They derive its exact gradient and prove its computational complexity is equivalent to that of hypervolume gradient computation. By employing hierarchical aggregation, the method establishes a connection to infinitesimal coding, thereby elucidating the underlying mechanics of nonsmooth optimization. The algorithm further incorporates projected finite differences, repulsion and stagnation-recovery strategies, and coordinate-projected geometric expansion techniques. Experimental results on standard two- and three-objective benchmarks, as well as curved and hyperspherical Pareto fronts, demonstrate that the proposed approach efficiently and robustly approximates the Pareto front.

This study addresses the challenge of efficiently generating and managing Pareto-optimal solution sets (SOS) in heterogeneous multi-task environments. It proposes an evolutionary multi-task optimization framework to construct compact, task-specific SOS repositories for real-world applications such as engineering design, inventory management, and hyperparameter optimization. The work introduces a novel similarity metric between Pareto sets and, for the first time, systematically validates the cross-domain applicability of SOS. Through visualization and objective space analysis, it reveals dynamic patterns in solution set performance across diverse task contexts. Experimental results demonstrate that the proposed approach effectively captures inter-task differences in solution sets and significantly enhances decision-making support across varying scenarios.

This work addresses the limitations of existing bi-objective optimization benchmark problems, which often suffer from distortion or uncontrolled complexity and lack test suites that are both realistic and theoretically tractable. To bridge this gap, the authors propose a construction method based on combinations of convex quadratic functions, enabling precise control over problem characteristics such as decision variable dimensionality, modality, and condition number. The resulting BONO-Bench suite comprises 20 problem classes with analytically traceable Pareto sets, controllable Pareto front shapes, and adjustable numbers of local optima. Notably, it is the first benchmark to unify theoretical solvability with flexible configuration of multidimensional attributes. The accompanying open-source Python package, bonobench, supports exact computation of hypervolume and R2 indicators, offering a high-fidelity, reproducible evaluation framework for multi-objective optimization algorithms.

This work addresses the inefficiency and suboptimality of manually tuned weight assignments in multi-objective reinforcement learning for industrial automation. It proposes the first integration of multi-objective Bayesian optimization with reinforcement learning, leveraging a sampling strategy based on the expected hypervolume improvement (qEHVI) to efficiently explore the Pareto front between energy consumption and control performance. Evaluated on the Quanser Aero 2 platform in a one-degree-of-freedom pitch control task, the method significantly outperforms uniform grid search by achieving superior hypervolume metrics and broader policy distribution coverage with substantially fewer evaluations. This demonstrates a sample-efficient approach to automatically discovering diverse Pareto-optimal policies without manual trade-off tuning.