Multi-Objective Optimization

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Overview

The field studies problems where improving one objective—such as accuracy, cost, safety, energy use, or fairness—can worsen another, and represents the alternatives as a Pareto frontier. Common approaches include evolutionary algorithms such as NSGA-II, Bayesian optimization for expensive evaluations, gradient-based methods for learning systems, and reinforcement-learning or generative methods for sequential and design tasks. Researchers also develop preference elicitation and constraint-handling methods so people can select useful trade-offs without always computing an entire frontier. A central tension is balancing broad, reliable coverage of feasible trade-offs against evaluation cost, while scalarization, objective scaling, and changing preferences can all affect which solutions are found.

Latest papers

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270 papers40 papers in the last four weeks, up 48% on the four weeks before. 0.2% of all new papers.

Mar 12, 2026

Few-for-Many Personalized Federated Learning

Personalized Federated Learning (PFL) aims to train customized models for clients with highly heterogeneous data distributions while preserving data privacy. Existing approaches often rely on heuristics like clustering or model interpolation, which lack principled mechanisms for balancing heterogeneous client objectives. Serving MM clients with distinct data distributions is inherently a multi-objective optimization problem, where achieving optimal personalization ideally requires MM distinct models on the Pareto front. However, maintaining MM separate models poses significant scalability challenges in federated settings with hundreds or thousands of clients. To address this challenge, we reformulate PFL as a few-for-many optimization problem that maintains only KK shared server models (K≪MK \ll M) to collectively serve all MM clients. We prove that this framework achieves near-optimal personalization: the approximation error diminishes as KK increases and each client's model converges to each client's optimum as data grows. Building on this reformulation, we propose FedFew, a practical algorithm that jointly optimizes the KK server models through efficient gradient-based updates. Unlike clustering-based approaches that require manual client partitioning or interpolation-based methods that demand careful hyperparameter tuning, FedFew automatically discovers the optimal model diversity through its optimization process. Experiments across vision, NLP, and real-world medical imaging datasets demonstrate that FedFew, with just 3 models, consistently outperforms other state-of-the-art approaches. Code is available at https://github.com/pgg3/FedFew.

Ping Guo, Tiantian Zhang, Xi Lin +3

City University of Hong Kong · CityU Shenzhen Research Institute · Hong Kong Metropolitan University +3

Federated LearningHeterogeneous Federated LearningMulti-Objective Optimization
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Mar 12, 2026

An Evolutionary Algorithm with Probabilistic Annealing for Large-scale Sparse Multi-objective Optimization

Large-scale sparse multi-objective optimization problems (LSMOPs) are prevalent in real-world applications, where optimal solutions typically contain only a few nonzero variables, such as in adversarial attacks, critical node detection, and sparse signal reconstruction. Since the function evaluation of LSMOPs often relies on large-scale datasets involving a large number of decision variables, the search space becomes extremely high-dimensional. The coexistence of sparsity and high dimensionality greatly intensifies the conflict between exploration and exploitation, making it difficult for existing multi-objective evolutionary algorithms (MOEAs) to identify the critical nonzero decision variables within limited function evaluations. To address this challenge, this paper proposes an evolutionary algorithm with probabilistic annealing for large-scale sparse multi-objective optimization. The algorithm is driven by two probability vectors with distinct entropy characteristics: a convergence-oriented probability vector with relatively low entropy ensures stable exploitation, whereas an annealed probability vector with gradually decreasing entropy enables an adaptive transition from global exploration to local refinement. By integrating these complementary search dynamics, the proposed algorithm achieves a dynamic equilibrium between exploration and exploitation. Experimental results on benchmark problems and real-world applications demonstrate that the proposed algorithm outperforms state-of-the-art evolutionary algorithms in terms of both convergence and diversity.

Shuai Shao, Yuhao Sun, Xing Chen +3

School of Computer Science and Technology, Anhui University, Hefei 230601, China · Sapient Intelligence, Singapore

Multi-Party Multi-Objective OptimizationMulti-Objective OptimizationMulti-Objective Evolutionary Optimization
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Mar 9, 2026

A Multi-Objective Optimization Approach for Sustainable AI-Driven Entrepreneurship in Resilient Economies

The rapid advancement of artificial intelligence (AI) technologies presents both unprecedented opportunities and significant challenges for sustainable economic development. While AI offers transformative potential for addressing environmental challenges and enhancing economic resilience, its deployment often involves substantial energy consumption and environmental costs. This research introduces the EcoAI-Resilience framework, a multi-objective optimization approach designed to maximize the sustainability benefits of AI deployment while minimizing environmental costs and enhancing economic resilience. The framework addresses three critical objectives through mathematical optimization: sustainability impact maximization, economic resilience enhancement, and environmental cost minimization. The methodology integrates diverse data sources, including energy consumption metrics, sustainability indicators, economic performance data, and entrepreneurship outcomes across 53 countries and 14 sectors from 2015-2024. Our experimental validation demonstrates exceptional performance with R scores exceeding 0.99 across all model components, significantly outperforming baseline methods, including Linear Regression (R = 0.943), Random Forest (R = 0.957), and Gradient Boosting (R = 0.989). The framework successfully identifies optimal AI deployment strategies featuring 100% renewable energy integration, 80% efficiency improvement targets, and optimal investment levels of $202.48 per capita. Key findings reveal strong correlations between economic complexity and resilience (r = 0.82), renewable energy adoption and sustainability outcomes (r = 0.71), and demonstrate significant temporal improvements in AI readiness (+1.12 points/year) and renewable energy adoption (+0.67 year) globally.

Anas ALsobeh, Raneem Alkurdi

Information Technology (ITEC), Southern Illinois University, Carbondale, USA · Business Information Technology Department, Yarmouk University, Irbid, Jordan

Multi-Objective Optimization
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Mar 9, 2026

First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.

Changkai Li

Constrained OptimizationMulti-Objective OptimizationSpectral Methods
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Mar 4, 2026

Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation

The use of stochastic differential equations in multi-objective optimization has been limited, in practice, by two persistent gaps: incomplete stability analyses and the absence of accessible implementations. We revisit a drift--diffusion model for unconstrained vector optimization in which the drift is induced by a common descent direction and the diffusion term preserves exploratory behavior. The main theoretical contribution is a self-contained Lyapunov analysis establishing global existence, pathwise uniqueness, and non-explosion under a dissipativity condition, together with positive recurrence under an additional coercivity assumption. We also derive an Euler--Maruyama discretization and implement the resulting iteration as a pymoo-compatible algorithm -- pymoo being an open-source Python framework for multi-objective optimization -- with an interactive PymooLab front-end for reproducible experiments. Empirical results on DTLZ2 with objective counts from three to fifteen indicate a consistent trade-off: compared with established evolutionary baselines, the method is less competitive in low-dimensional regimes but remains a viable option under restricted evaluation budgets in higher-dimensional settings. Taken together, these observations suggest that stochastic drift--diffusion search occupies a mathematically tractable niche alongside population-based heuristics -- not as a replacement, but as an alternative whose favorable properties are amenable to rigorous analysis.

Thiago Santos, Sebastiao Xavier

Department of Mathematics, Federal University of Ouro Preto, UFOP, Ouro Preto, Brazil

Multi-Objective OptimizationStochastic Differential EquationsDynamical Systems
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Mar 4, 2026

An Adaptive KKT-Based Indicator for Convergence Assessment in Multi-Objective Optimization

Performance indicators are essential tools for assessing the convergence behavior of multi-objective optimization algorithms, particularly when the true Pareto front is unknown or difficult to approximate. Classical reference-based metrics such as hypervolume and inverted generational distance are widely used, but may suffer from scalability limitations and sensitivity to parameter choices in many-objective scenarios. Indicators derived from Karush--Kuhn--Tucker (KKT) optimality conditions provide an intrinsic alternative by quantifying stationarity without relying on external reference sets. This paper revisits an entropy-inspired KKT-based convergence indicator and proposes a robust adaptive reformulation based on quantile normalization. The proposed indicator preserves the stationarity-based interpretation of the original formulation while improving robustness to heterogeneous distributions of stationarity residuals, a recurring issue in many-objective optimization.

Thiago Santos, Sebastiao Xavier

Department of Mathematics, Federal University of Ouro Preto, UFOP, Ouro Preto, Brazil

Multi-Objective Optimization
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Feb 27, 2026

Adaptive Combinatorial Experimental Design: Pareto Optimality for Decision-Making and Inference

In this paper, we provide the first investigation into adaptive combinatorial experimental design, focusing on the trade-off between regret minimization and statistical power in combinatorial multi-armed bandits (CMAB). While minimizing regret requires repeated exploitation of high-reward arms, accurate inference on reward gaps requires sufficient exploration of suboptimal actions. We formalize this trade-off through the concept of Pareto optimality and establish equivalent conditions for Pareto-efficient learning in CMAB. We consider two relevant cases under different information structures, i.e., full-bandit feedback and semi-bandit feedback, and propose two algorithms MixCombKL and MixCombUCB respectively for these two cases. We provide theoretical guarantees showing that both algorithms are Pareto optimal, achieving finite-time guarantees on both regret and estimation error of arm gaps. Our results further reveal that richer feedback significantly tightens the attainable Pareto frontier, with the primary gains arising from improved estimation accuracy under our proposed methods. Taken together, these findings establish a principled framework for adaptive combinatorial experimentation in multi-objective decision-making.

Hongrui Xie, Junyu Cao, Kan Xu

University of Science and Technology of China · The University of Texas at Austin · Arizona State University

Combinatorial Multi-Armed BanditsExploration-Exploitation TradeoffPareto OptimizationMulti-Objective Optimization
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Feb 25, 2026

Routing by Reaching: Composition of Pre-trained GFlowNets for Multi-Objective Generation

Generative Flow Networks (GFlowNets) learn to sample diverse candidates in proportion to a reward function, making them well-suited for scientific discovery, where exploring multiple promising solutions is crucial. Further extending GFlowNets to multi-objective settings has attracted growing interest as real-world applications often involve multiple, conflicting objectives. However, existing approaches require joint training for each combination of objectives, meaning that any change in the objective set necessitates retraining from scratch. We propose a framework that composes pre-trained GFlowNets at inference time, enabling rapid adaptation without fine-tuning or retraining. Importantly, our framework is flexible, capable of handling diverse reward combinations ranging from linear scalarization to complex nonlinear operators, which are often handled separately in previous literature. We prove that our method exactly recovers the target distribution for linear scalarization, and quantify the approximation quality for nonlinear operators through a distortion factor. Experiments on a synthetic 2D grid and real-world molecule generation tasks demonstrate that our approach achieves performance comparable to baselines.

Seokwon Yoon, Youngbin Choi, Seunghyuk Cho +3

Department of Computer Science & Engineering, POSTECH, South Korea · Graduate School of Artificial Intelligence, POSTECH, South Korea

Generative Flow NetworksGenerative ModelingMulti-Objective Optimization
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Feb 22, 2026

Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems

Fuzzy systems show strong potential in explainable AI due to their rule-based architecture and linguistic variables. Existing approaches navigate the accuracy-explainability trade-off either through evolutionary multi-objective optimization (MOO), which is computationally expensive, or gradient-based scalarization, which cannot recover non-convex Pareto regions. We propose X-ANFIS, an alternating bi-objective gradient-based optimization scheme for explainable adaptive neuro-fuzzy inference systems. Cauchy membership functions are used for stable training under semantically controlled initializations, and a differentiable explainability objective is introduced and decoupled from the performance objective through alternating gradient passes. Validated in approximately 5,000 experiments on nine UCI regression datasets, X-ANFIS consistently achieves target distinguishability while maintaining competitive predictive accuracy, recovering solutions beyond the convex hull of the MOO Pareto front.

Qusai Khaled, Uzay Kaymak, Laura Genga

Jheronimus Academy of Data Science Eindhoven University of Technology Eindhoven, The Netherlands · School of Industrial Engineering Eindhoven University of Technology Eindhoven, The Netherlands

Explainable Artificial IntelligenceFuzzy LogicMulti-Objective Optimization
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Feb 21, 2026

CaliCausalRank: Calibrated Multi-Objective Ad Ranking with Robust Counterfactual Utility Optimization

Ad ranking systems must simultaneously optimize multiple objectives including click-through rate (CTR), conversion rate (CVR), revenue, and user experience metrics. However, production systems face critical challenges: score scale inconsistency across traffic segments undermines threshold transferability, and position bias in click logs causes offline-online metric discrepancies. We propose CaliCausalRank, a unified framework that integrates training-time scale calibration, constraint-based multi-objective optimization, and robust counterfactual utility estimation. Our approach treats score calibration as a first-class training objective rather than post-hoc processing, employs Lagrangian relaxation for constraint satisfaction, and utilizes variance-reduced counterfactual estimators for reliable offline evaluation. Experiments on the Criteo and Avazu datasets demonstrate that CaliCausalRank achieves 1.1% relative AUC improvement, 31.6% calibration error reduction, and 3.2% utility gain compared to the best baseline (PairRank) while maintaining consistent performance across different traffic segments.

Xikai Yang, Sebastian Sun, Yilin Li +3

Columbia University, New York, USA · University of Wisconsin-Madison, Madison, USA · Carnegie Mellon University, Pittsburgh, USA +3

Learning to RankMulti-Objective OptimizationCTR Prediction
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Feb 18, 2026

End-user validation of BRIGHT with custom-developed graphical user interface applied to cervical cancer brachytherapy

Multi-objective optimisation using BRIGHT has proven insightful and effective in prostate cancer brachytherapy treatment planning. BRachytherapy via artificially Intelligent GOMEA-Heuristic based Treatment planning (BRIGHT) generates multiple treatment plans, each with a different trade-off between tumour coverage and organs-at-risk sparing. BRIGHT was recently extended to cervical cancer brachytherapy. In this study, we present a novel, custom-developed graphical user interface (GUI) that enables plan navigation, pairwise comparisons, dose distribution visualisation, and possibility for adjustments - essential for efficient clinical use of BRIGHT. End-user validation of BRIGHT with the dedicated GUI was conducted for cervical cancer brachytherapy by emulating clinical practice in ten previously treated patients. A multidisciplinary brachytherapy team used BRIGHT to create new treatment plans. GUI usability was assessed using the System Usability Scale (SUS). BRIGHT plan quality was compared to clinical practice via blinded one-on-one comparisons. The GUI offered helpful features for plan navigation and evaluation, giving users quick insight into whether planning aims are achievable and what treatment options are available. The overall SUS score was 83.3, indicating an 'excellent' system. BRIGHT outperformed clinical practice in five out of ten patients regarding the coverage-sparing trade-off and performed equally well in the remaining five. The BRIGHT plan was preferred over the clinical plan in eight out of ten patients, four of which showed clinically relevant differences. The clinical plan was preferred in two patients, neither with clinically relevant differences. In conclusion, BRIGHT, with its dedicated GUI, is a clinically viable and user-friendly tool for treatment planning in cervical cancer brachytherapy.

Leah R. M. Dickhoff, Ellen M. Kerkhof, Heloisa H. Deuzeman +12

OncologyClinical Decision SupportMulti-Objective Optimization
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Feb 17, 2026

AI-CARE: Carbon-Aware Reporting Evaluation Metric for AI Models

As machine learning (ML) continues its rapid expansion, the environmental cost of model training and inference has become a critical societal concern. Existing benchmarks overwhelmingly focus on standard performance metrics such as accuracy, BLEU, or mAP, while largely ignoring energy consumption and carbon emissions. This single-objective evaluation paradigm is increasingly misaligned with the practical requirements of large-scale deployment, particularly in energy-constrained environments such as mobile devices, developing regions, and climate-aware enterprises. In this paper, we propose AI-CARE, an evaluation tool for reporting energy consumption, and carbon emissions of ML models. In addition, we introduce the carbon-performance tradeoff curve, an interpretable tool that visualizes the Pareto frontier between performance and carbon cost. We demonstrate, through theoretical analysis and empirical validation on representative ML workloads, that carbon-aware benchmarking changes the relative ranking of models and encourages architectures that are simultaneously accurate and environmentally responsible. Our proposal aims to shift the research community toward transparent, multi-objective evaluation and align ML progress with global sustainability goals. The tool and documentation are available at https://github.com/USD-AI-ResearchLab/ai-care.

KC Santosh, Srikanth Baride, Rodrigue Rizk

USD Artificial Intelligence Research, Department of Computer Science University of South Dakota Vermillion, SD 57069, USA

Carbon-Aware ComputingEnergy-Efficient MLMulti-Objective Optimization
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Feb 17, 2026

DNN-Enabled Multi-User Beamforming for Throughput Maximization under Adjustable Fairness

Ensuring user fairness in wireless communications is a fundamental challenge, as balancing the trade-off between fairness and sum rate leads to a non-convex, multi-objective optimization whose complexity grows with network scale. To alleviate this conflict, we propose an optimization-based unsupervised learning approach based on the wireless transformer (WiT) architecture that learns from channel state information (CSI) features. We reformulate the trade-off by combining the sum rate and fairness objectives through a Lagrangian multiplier, which is updated automatically via a dual-ascent algorithm. This mechanism allows for a controllable fairness constraint while simultaneously maximizing the sum rate, effectively realizing a trace on the Pareto front between two conflicting objectives. Our findings show that the proposed approach offers a flexible solution for managing the trade-off optimization under prescribed fairness.

Kaifeng Lu, Markus Rupp, Stefan Schwarz

Institute of Telecommunications, Technische Universität (TU) Wien

Wireless Resource AllocationMulti-Objective Optimization
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Feb 16, 2026

Locally Adaptive Multi-Objective Learning

We consider the general problem of learning a predictor that satisfies multiple objectives of interest simultaneously, a broad framework that captures a range of specific learning goals including calibration, regret, and multiaccuracy. We work in an online setting where the data distribution can change arbitrarily over time. Existing approaches to this problem aim to minimize the set of objectives over the entire time horizon in a worst-case sense, and in practice they do not necessarily adapt to distribution shifts. Earlier work has aimed to alleviate this problem by incorporating additional objectives that target local guarantees over contiguous subintervals. Empirical evaluation of these proposals is, however, scarce. In this article, we consider an alternative procedure that achieves local adaptivity by replacing one part of the multi-objective learning method with an adaptive online algorithm. Empirical evaluations on datasets from energy forecasting and algorithmic fairness show that our proposed method improves upon existing approaches and achieves unbiased predictions over subgroups, while remaining robust under distribution shift.

Jivat Neet Kaur, Isaac Gibbs, Michael I. Jordan

University of California, Berkeley · Inria, Paris

Multi-Objective OptimizationAdaptive Learning
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Feb 16, 2026

MAC-AMP: A Closed-Loop Multi-Agent Collaboration System for Multi-Objective Antimicrobial Peptide Design

To address the global health threat of antimicrobial resistance, antimicrobial peptides (AMP) are being explored for their potent and promising ability to fight resistant pathogens. While artificial intelligence (AI) is being employed to advance AMP discovery and design, most AMP design models struggle to balance key goals like activity, toxicity, and novelty, using rigid or unclear scoring methods that make results hard to interpret and optimize. As the capabilities of Large Language Models (LLM) advance and evolve swiftly, we turn to AI multi-agent collaboration based on such models (multi-agent LLMs), which show rapidly rising potential in complex scientific design scenarios. Based on this, we introduce MAC-AMP, a closed-loop multi-agent collaboration (MAC) system for multi-objective AMP design. The system implements a fully autonomous simulated peer review-adaptive reinforcement learning framework that requires only a task description and example dataset to design novel AMPs. The novelty of our work lies in introducing a closed-loop multi-agent system for AMP design, with cross-domain transferability, that supports multi-objective optimization while remaining explainable rather than a 'black box'. Experiments show that MAC-AMP outperforms other AMP generative models by effectively optimizing AMP generation for multiple key molecular properties, demonstrating exceptional results in antibacterial activity, AMP likeliness, toxicity compliance, and structural reliability.

Gen Zhou, Sugitha Janarthanan, Lianghong Chen +1

Department of Computer Science, Western University, London, ON, Canada · Department of Biochemistry, Western University, London, ON, Canada

Molecular GenerationMulti-Agent LLM SystemsMulti-Agent CollaborationMulti-Objective Optimization
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Feb 8, 2026

Multi-Agent Route Planning as a QUBO Problem

Multi-Agent Route Planning considers selecting vehicles, each associated with a single predefined route, such that route-level coverage utility is maximized while redundant spatial overlaps are limited. This paper gives a formal problem definition, proves NP-hardness by reduction from the Weighted Set Packing problem, and derives a Quadratic Unconstrained Binary Optimization formulation whose coefficients directly encode route utility rewards and pairwise overlap penalties. A single penalty parameter λλ controls the coverage--overlap trade-off. We distinguish between a soft regime, which supports multi-objective exploration, and a hard regime, in which the penalty is strong enough to effectively enforce near-disjoint routes. We describe a practical pipeline for generating city instances, constructing candidate routes, building the QUBO matrix, and solving it with a binary quadratic programming baseline (Gurobi), simulated annealing, and D-Wave hybrid quantum annealing. Experiments on Barcelona instances with up to 10,00010{,}000 vehicles reveal a clear coverage--overlap knee and show that Pareto-optimal solutions are mainly obtained under the hard-penalty regime, while D-Wave hybrid solvers and Gurobi achieve very similar objective values on matching configurations with only minor runtime differences as problem size grows.

Renáta Rusnáková, Martin Chovanec, Juraj Gazda

Department of Computers and Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice

Multi-Objective Optimization
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Feb 6, 2026

Uncovering Cross-Objective Interference in Multi-Objective Alignment

We study a persistent failure mode in multi-objective alignment for large language models (LLMs), in which scalarized training improves only some objectives while the others degrade. We formalize this phenomenon as cross-objective interference and, to our knowledge, conduct the first systematic study of scalarization algorithms for multi-objective LLM alignment. The study shows that interference is pervasive across algorithms yet strongly model-dependent. To understand how interference arises, we derive a local covariance law stating that an objective improves or degrades at first order according to the sign of the covariance between its reward and the scalarized score. We extend this law to the clipped surrogate objectives of modern reinforcement fine-tuning and show that it still holds under mild conditions. Building on this law, we propose COVariance-floor Enforced Reweighting (COVER), a one-sided controller that raises an objective's weight only when the covariance between its reward and the clipped advantage weight falls below a target. Through extensive experiments, we find that COVER can mitigate cross-objective interference while matching linear scalarization when objectives already co-improve. Finally, to explain why interference is model-dependent, we complement the local covariance law with a global convergence analysis. This analysis gives sufficient conditions for the non-convex scalarized objective to satisfy the Polyak--Łojasiewicz condition and relates interference to model geometry.

Yining Lu, Meng Jiang

University of Notre Dame

Adaptive Loss WeightingMulti-Objective Language Model AlignmentMulti-Objective Optimization
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Feb 5, 2026

Variable Search Stepsize for Randomized Local Search in Multi-Objective Combinatorial Optimization

Over the past two decades, research in evolutionary multi-objective optimization has predominantly focused on continuous domains, with comparatively limited attention given to multi-objective combinatorial optimization problems (MOCOPs). Combinatorial problems differ significantly from continuous ones in terms of problem structure and landscape. Recent studies have shown that on MOCOPs multi-objective evolutionary algorithms (MOEAs) can even be outperformed by simple randomised local search. Starting with a randomly sampled solution in search space, randomised local search iteratively draws a random solution (from an archive) to perform local variation within its neighbourhood. However, in most existing methods, the local variation relies on a fixed neighbourhood, which limits exploration and makes the search easy to get trapped in local optima. In this paper, we present a simple yet effective local search method, called variable stepsize randomized local search (VS-RLS), which adjusts the stepsize during the search. VS-RLS transitions gradually from a broad, exploratory search in the early phases to a more focused, fine-grained search as the search progresses. We demonstrate the effectiveness and generalizability of VS-RLS through extensive evaluations against local search and MOEAs methods on diverse MOCOPs.

Xuepeng Ren, Maocai Wang, Guangming Dai +4

School of Computer, China University of Geosciences, Engineering Research Center of Natural Resource Information Management and Digital Twin Engineering Software, Ministry of Education, and Hubei Key Laboratory of Intelligent Geo-Information Processing, Wuhan 430074, China · School of Computer, Science and Informatics, De Montfort University, Leicester LE1 9BH, U.K. · School of Computer Science, University of Birmingham, Birmingham B15 2TT, U.K. +1

Metaheuristic OptimizationMulti-Objective Optimization
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Feb 3, 2026

NeuroPareto: Calibrated Acquisition for Costly Many-Goal Search in Vast Parameter Spaces

The pursuit of optimal trade-offs in high-dimensional search spaces under stringent computational constraints poses a fundamental challenge for contemporary multi-objective optimization. We develop NeuroPareto, a cohesive architecture that integrates rank-centric filtering, uncertainty disentanglement, and history-conditioned acquisition strategies to navigate complex objective landscapes. A calibrated Bayesian classifier estimates epistemic uncertainty across non-domination tiers, enabling rapid generation of high-quality candidates with minimal evaluation cost. Deep Gaussian Process surrogates further separate predictive uncertainty into reducible and irreducible components, providing refined predictive means and risk-aware signals for downstream selection. A lightweight acquisition network, trained online from historical hypervolume improvements, guides expensive evaluations toward regions balancing convergence and diversity. With hierarchical screening and amortized surrogate updates, the method maintains accuracy while keeping computational overhead low. Experiments on DTLZ and ZDT suites and a subsurface energy extraction task show that NeuroPareto consistently outperforms classifier-enhanced and surrogate-assisted baselines in Pareto proximity and hypervolume.

Rong Fu, Chunlei Meng, Haoyu Zhao +2

Independent Researcher

Multi-Objective OptimizationSurrogate-Assisted OptimizationBayesian OptimizationUncertainty-Aware Exploration
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Jan 30, 2026

Fairness-Aware Performance Evaluation for Multi-Party Multi-Objective Optimization

In multiparty multiobjective optimization problems, solution sets are usually evaluated using classical performance metrics, aggregated across DMs. However, such mean-based evaluations may be unfair by favoring certain parties, as they assume identical geometric approximation quality to each party's PF carries comparable evaluative significance. Moreover, prevailing notions of MPMOP optimal solutions are restricted to strictly common Pareto optimal solutions, representing a narrow form of cooperation in multiparty decision making scenarios. These limitations obscure whether a solution set reflects balanced relative gains or meaningful consensus among heterogeneous DMs. To address these issues, this paper develops a fairness-aware performance evaluation framework grounded in a generalized notion of consensus solutions. From a cooperative game-theoretic perspective, we formalize four axioms that a fairness-aware evaluation function for MPMOPs should satisfy. By introducing a concession rate vector to quantify acceptable compromises by individual DMs, we generalize the classical definition of MPMOP optimal solutions and embed classical performance metrics into a Nash-product-based evaluation framework, which is theoretically shown to satisfy all axioms. To support empirical validation, we further construct benchmark problems that extend existing MPMOP suites by incorporating consensus-deficient negotiation structures. Experimental results demonstrate that the proposed evaluation framework is able to distinguish algorithmic performance in a manner consistent with consensus-aware fairness considerations. Specifically, algorithms converging toward strictly common solutions are assigned higher evaluation scores when such solutions exist, whereas in the absence of strictly common solutions, algorithms that effectively cover the commonly acceptable region are more favorably evaluated.

Zifan Zhao, Peilan Xu, Wenjian Luo

School of Artificial Intelligence, Nanjing University of Information Science and Technology, Nanjing 210044, China · Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies, Institute of Cyberspace Security, School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen 518055, China

Multi-Party Multi-Objective OptimizationCooperative Game TheoryAlgorithmic FairnessMulti-Objective Optimization
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Dec 30, 2025

Decoupling Constraints from Two Directions for Evolutionary Constrained Multi-objective Optimization

Real-world constrained multi-objective optimization problems (CMOPs) commonly involve multiple constraints, and understanding and exploiting their coupling relationships is crucial for efficient optimization. Recent constraint-decoupling methods handle individual constraints separately, but they generally search only in the evolutionary direction to approximate single-constraint Pareto fronts (SCPFs). In this study, we show that part or all of the constrained Pareto front (CPF) may be unrelated to any SCPF and instead be shaped by the boundaries of infeasible regions. We refer to such a portion as the independent CPF (ICPF) and introduce the reverse CPF (RCPF) to characterize its associated informative infeasible boundaries. Based on these observations, we propose a bidirectional constraint-decoupling coevolutionary algorithm named DCF2D. DCF2D dynamically identifies the constraints obstructing the main population and activates constraint-specific auxiliary populations. These populations adaptively search in the evolutionary direction for the corresponding SCPFs or in the reverse evolutionary direction for the corresponding RCPFs. Its three-stage framework integrates unconstrained global exploration, event-driven bidirectional coevolution, and final convergence refinement. Experiments on 87 benchmark instances from seven test suites and 28 real-world engineering CMOPs demonstrate that DCF2D achieves the best overall performance among nine algorithms. Code available at: https://github.com/RuiqingS/DCF2D.

Ruiqing Sun, Dawei Feng, Xing Zhou +6

College of Computer Science and Technology, National University of Defense Technology, Changsha 410000, P.R. China · College of Intelligence Science and Technology, National University of Defense Technology, Changsha, 410000, P.R. China · State Key Laboratory of Complex & Critical Software Environment, College of Information and Communication, National University of Defense Technology, Wuhan 430019, P.R. China +1

Evolutionary OptimizationMulti-Objective Evolutionary OptimizationConstrained OptimizationMulti-Objective Optimization
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Dec 20, 2025

Large-scale benchmarking of multi-objective soft-computing metaheuristics for redundancy allocation in repairable k-out-of-n systems

This paper presents a large-scale budget-aware benchmark of multi-objective soft-computing metaheuristics for a bi-objective redundancy allocation problem in repairable k-out-of-n systems. The problem combines cost minimization and steady-state availability maximization under weight constraints, with binary subsystem-level decisions determining both the number of redundant components and the redundancy strategy. Four strategies are considered: cold standby, warm standby, hot standby, and a mixed active-warm standby strategy. Subsystem availability is evaluated using continuous-time Markov chains, while the optimization task is treated as a constrained mixed-integer multi-objective problem. The study compares 65 metaheuristic algorithms from multiple algorithmic families under two initialization settings, with and without Scaled Binomial Initialization (SBI), across six case studies of increasing structural and dimensional complexity and four weight limits per case. Performance is assessed using hypervolume, budget-dependent convergence profiles, and non-parametric statistical comparisons. The results show that algorithm rankings are strongly budget-dependent, indicating that a single final-budget ranking can be misleading. SBI provides a substantial early advantage and can change the relative performance of competing methods, especially for larger instances. The best-performing algorithms vary across budget regimes: NNIA-SBI and CMOPSO-SBI are competitive under tight budgets, whereas NSGA-II+ARSBX-SBI performs robustly for medium and large budgets. From a system design perspective, Pareto-optimal solutions are dominated by hot standby and mixed redundancy strategies, while cold and warm standby rarely appear. The benchmark highlights the importance of initialization, computational budget, and problem complexity when selecting soft-computing optimizers for practical redundancy allocation.

Mateusz Oszczypała, David Ibehej, Jakub Kudela

Military University of Technology, gen. Sylwestra Kaliskiego Street 2, 00-908 Warsaw, Poland. · Institute of Automation and Computer Science, Brno University of Technology, Brno, Czech Republic.

Benchmark DesignMetaheuristic OptimizationMulti-Objective Optimization
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Nov 9, 2025

Feature weighting for data analysis via evolutionary simulation

We analyze an algorithm for assigning weights prior to scalarization in discrete multi-objective problems arising from data analysis. The algorithm evolves weights (interpreted as the relevance of features) by a replicator-type dynamic on the standard simplex, with update indices computed from a normalized data matrix. We prove that the resulting sequence converges globally to a unique interior equilibrium, yielding non-degenerate limiting weights.

Aris Daniilidis, Alberto Domínguez Corella, Philipp Wissgott

Institut für Stochastik und Wirtschaftsmathematik, Variational Analysis, Dynamics and Operations Research Unit E105-04, Technische Universität Wien, Wiedner Hauptstraße 8, 1040 Vienna, Austria · Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Heinrichstraße 36, A-8010 Graz, Austria · danube.ai solutions gmbh, 1040 Vienna, Austria

Replicator DynamicsImportance WeightingMulti-Objective Optimization
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Sep 26, 2025

OFMU: Optimization-Driven Framework for Machine Unlearning

Large language models deployed in sensitive applications increasingly require the ability to unlearn specific knowledge, such as user requests, copyrighted materials, or outdated information, without retraining from scratch to ensure regulatory compliance, user privacy, and safety. This task, known as machine unlearning, aims to remove the influence of targeted data (forgetting) while maintaining performance on the remaining data (retention). A common approach is to formulate this as a multi-objective problem and reduce it to a single-objective problem via scalarization, where forgetting and retention losses are combined using a weighted sum. However, this often results in unstable training dynamics and degraded model utility due to conflicting gradient directions. To address these challenges, we propose OFMU, a penalty-based bi-level optimization framework that explicitly prioritizes forgetting while preserving retention through a hierarchical structure. Our method enforces forgetting via an inner maximization step that incorporates a similarity-aware penalty to decorrelate the gradients of the forget and retention objectives, and restores utility through an outer minimization step. To ensure scalability, we develop a two-loop algorithm with provable convergence guarantees under both convex and non-convex regimes. We further provide a rigorous theoretical analysis of convergence rates and show that our approach achieves better trade-offs between forgetting efficacy and model utility compared to prior methods. Extensive experiments across vision and language benchmarks demonstrate that OFMU consistently outperforms existing unlearning methods in both forgetting efficacy and retained utility.

Sadia Asif, Mohammad Mohammadi Amiri

Rensselaer Polytechnic Institute

Bilevel OptimizationMachine UnlearningMulti-Objective OptimizationLLM Unlearning
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Jun 27, 2025

Interactive Multi-Objective Probabilistic Preference Learning with Soft and Hard Bounds

High-stakes decision-making involves navigating multiple competing objectives with expensive evaluations. For instance, in brachytherapy, clinicians must balance maximizing tumor coverage (e.g., an aspirational target or soft bound of >95% coverage) against strict organ dose limits (e.g., a non-negotiable hard bound of <601cGy to the bladder). Selecting Pareto-optimal solutions that match implicit preferences is challenging, as exhaustive Pareto frontier exploration is computationally and cognitively prohibitive, necessitating interactive frameworks to guide users. While decision-makers (DMs) often possess domain knowledge to narrow the search via such soft-hard bounds, current methods often lack systematic approaches to iteratively refine these multi-faceted preference structures. Furthermore, DMs often require confidence that they have not overlooked superior alternatives, a paramount necessity in high-stakes scenarios. We present Active-MoSH, an interactive local-global framework designed for this process. Its local component integrates probabilistic preference learning with an active sampling strategy to adaptively refine Pareto subsets while minimizing cognitive burden. To bolster decision confidence, Active-MoSH's global component, C-MoSH, leverages multi-objective sensitivity analysis to identify potentially overlooked, high-value points beyond immediate feedback. We demonstrate Active-MoSH's performance benefits through diverse synthetic and real-world applications. A high-stakes case study with real cervical cancer brachytherapy treatment plans and an image selection user study further validate our hypotheses regarding the framework's ability to improve convergence, enhance DM confidence, and provide expressive preference articulation.

Edward Chen, Sang T. Truong, Natalie Dullerud +2

Department of Computer Science, Stanford University, Stanford, California, USA

Active LearningMulti-Criteria Decision AnalysisMulti-Objective OptimizationPreference Learning
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Dec 11, 2024

Variational Model Merging for Pareto Front Estimation in Multitask Finetuning

Pareto fronts are useful to find good task-mixing strategies for multitask finetuning, but they are also costly to compute. To reduce costs, recent works have used existing model merging methods to help train cheap surrogate models to estimate the Pareto fronts. However, no work has yet considered designing new model-merging methods to directly, and provably, improve the quality of Pareto fronts. Here, we fill this gap by proposing a new Bayesian approach called Variational Model Merging. In this approach, existing model-merging methods are obtained as special cases of "posterior-merging" when Gaussian posteriors are used and new model-merging strategies can be derived by using non-Gaussian posteriors. Our main theoretical result is to show that more flexible posteriors necessarily yield better estimates of Pareto fronts. For instance, a Pareto front estimate obtained by merging full-Gaussian posteriors is expected to be better than that obtained by using isotropic Gaussian posteriors. We validate the theory through extensive empirical results on vision and language transformers where better Gaussian families consistently yields better or comparable Pareto fronts. Our work is a rare instance where Bayesian ideas are used to improve Pareto analysis.

Hugo Monzón Maldonado, Nico Daheim, Thomas Möllenhoff +2

RIKEN Center for AI Project, Japan · Ubiquitous Knowledge Processing Lab (UKP Lab), Department of Computer Science, Technical University of Darmstadt and National Research Center for Applied Cybersecurity ATHENE, Germany · National Research Center for Applied Cybersecurity ATHENE, Germany

Bayesian InferenceMulti-Task LearningMulti-Task OptimizationLanguage Model MergingVariational InferenceMulti-Objective Optimization
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Sep 17, 2024

Lexicographic Descent: A Gradient-Based Approach to Optimize Totally Ordered Rulebooks via Parametric Scalarization

Safety-critical systems, such as autonomous vehicles, must satisfy conflicting constraints and objectives, collectively called rules, typically ordered lexicographically. Preemptive lexicographic optimization solves a sequence of problems, one per priority level. However, higher levels are optimized without considering lower ones, making the approach susceptible to suboptimal solutions. On the other hand, scalarization optimizes all levels jointly and so avoids this myopia, but a finite weighted sum permits trade-offs across priority levels, and exact scalar representations lack the regularity standard solvers require. We propose Lexicographic Descent, which combines the benefits of both: a single weighted sum whose relative weights decay at priority-dependent rates, recovering the lexicographic order in the limit. We show that Lexicographic Descent converges to constrained stationary points of the highest-priority rule, matching preemptive guarantees. Since the lower-priority weights vanish, we derive a necessary condition for the lower-priority rules to keep guiding the iterates, with the aim of avoiding suboptimal stationary points of the higher-priority rules. We show that Lexicographic Descent can exploit problem structure to reach the lexicographic optima, and outperforms the preemptive approach on an autonomous driving case study.

Matteo Penlington, Alessandro Zanardi, Lorenzo Fiaschi +1

Institute for Dynamic Systems and Control, ETH Zürich, Switzerland

Constrained OptimizationMulti-Objective Optimization
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Jun 23, 2024

Jacobian Descent for Multi-Objective Optimization

Many optimization problems require balancing multiple conflicting objectives. To generalize gradient descent, which is limited to single-objective optimization, we introduce Jacobian descent (JD) and its stochastic variants. This algorithm iteratively updates parameters using the Jacobian matrix of a vector-valued objective function, in which each row is the gradient of an individual objective. While several methods to combine gradients already exist in the literature, they generally struggle when the objectives conflict. In contrast, we propose projecting gradients to fully resolve conflict. We prove convergence to the Pareto front in the smooth convex case with this approach, a stronger guarantee than the usual convergence to a weakly Pareto stationary point. Our method also enables instance-wise risk minimization (IWRM), a novel learning paradigm in which the loss of each training example is considered a separate objective. Early experimentation on image classification shows promising results for IWRM. Lastly, we provide an efficient implementation of JD using the Gramian of the Jacobian matrix to drastically reduce memory requirements.

Pierre Quinton, Valérian Rey

Independent

Multi-Objective OptimizationPareto OptimizationGradient Descent
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Jan 7, 2024

Multi-Objective Hyperparameter Search via Damped Gauss--Newton Optimization

We study hyperparameter optimization (HPO) from a numerical-optimization perspective and propose a multi-objective damped Newton--Gauss--Newton search method. Rather than perturbing each hyperparameter separately or treating model evaluations as independent trials, the method uses performance changes between successive full hyperparameter vectors to construct an iterative secant approximation of the local sensitivity matrix. Each iteration therefore requires only one new full-vector model evaluation while jointly updating all optimized hyperparameters. A Tikhonov-regularized Gauss--Newton system stabilizes the update when the number of hyperparameters exceeds the number of performance objectives. The search is initialized from readily available empirical/default settings of the underlying learner, without requiring a separate initialization search. We evaluate four-dimensional XGBoost HPO on three public classification datasets against exhaustive grid search, random search, and tree-structured Parzen estimator (TPE) optimization. On Breast Cancer Wisconsin, the proposed method matches the best validation accuracy of a 320-configuration grid search with slightly better log loss and ROC--AUC. Across three datasets and three seeds, predictive performance remains competitive with random search and TPE while using substantially fewer search iterations. A supplementary proprietary eight-dimensional threshold-optimization case study further demonstrates joint multi-parameter optimization under competing performance targets and reveals non-monotonic, oscillatory trajectories that motivate damping and best-iterate selection. Overall, the results establish iterative secant sensitivity as an evaluation-efficient local alternative to exhaustive HPO.

Qinwu Xu, Yifan Jiang

Gauss-Newton OptimizationSecond-Order OptimizationMulti-Objective OptimizationHyperparameter Optimization
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May 15, 2023

Python Tool for Visualizing Variability of Pareto Fronts over Multiple Runs

Hyperparameter optimization is crucial to achieving high performance in deep learning. On top of the performance, other criteria such as inference time or memory requirement often need to be optimized due to some practical reasons. This motivates research on multi-objective optimization (MOO). However, Pareto fronts of MOO methods are often shown without considering the variability caused by random seeds and this makes the performance stability evaluation difficult. Although there is a concept named empirical attainment surface to enable the visualization with uncertainty over multiple runs, there is no major Python package for empirical attainment surface. We, therefore, develop a Python package for this purpose and describe the usage. The package is available at https://github.com/nabenabe0928/empirical-attainment-func. OptunaHub now provides our implementation at https://hub.optuna.org/visualization/plot_empirical_attainment_surface/.

Shuhei Watanabe

Department of Computer Science, University of Freiburg, Germany

Multi-Objective OptimizationPareto Optimization
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