Pareto Optimization

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Oct 5, 2026cs.LG

Learning Pareto Stationary Fronts via Single-Pass Backpropagation

We propose MOSEL (Multi-Objective Stackelberg Efficient Learning), a framework for a posteriori multi-objective optimization (MOO) in deep neural networks that recovers a full front of Pareto stationary solutions at the computational cost of standard single-objective training. MOSEL reformulates the problem as a bilevel optimization problem that leverages network modularity to decouple representation learning from objective-preference alignment. Casting the bilevel optimization problem as a Stackelberg game enables solving the original a posteriori MOO problem in a single forward-backward pass. As a result, MOSEL matches the time and memory efficiency of standard single-objective training while enabling scalable Pareto stationary front learning. Empirically, MOSEL uncovers diverse and optimal Pareto frontiers in strongly conflicting settings (e.g., fairness-accuracy). Remarkably, even in weakly conflicting regimes such as multi-task learning, it consistently converges to solutions closer to the utopia point, outperforming both standard single-objective training and specialized multi-task learning methods. These results highlight the broader potential of a posteriori MOO learning as a pathway to efficiently learn more diverse and robust representations, ultimately improving generalization.

Elina Rojin Celik, Marcos Medeiros Raimundo, Isabel Valera

Department of Computer Science Saarland University, Germany · Instituto de Computação Universidade Estadual de Campinas, Brazil

Deep Learning Optimization Multi-Objective Optimization Pareto Optimization

Sep 7, 2026cs.LG

ParetoTransport: Generative Optimization by Mass Transport Toward The Pareto Front

Offline multi-objective optimization requires not only moving the objective vectors of candidate designs toward the Pareto front, but also distributing them effectively along it. Generative methods have recently emerged as a natural approach because they learn a distribution over feasible designs while allowing generation to be steered toward promising designs. Existing methods, however, largely retain classical sample-wise guidance strategies, leaving the distribution-level modeling capability of generative methods underused. We propose ParetoTransport, a training-free guidance method for pre-trained flow-matching models that explicitly specifies and refines a population-level distribution in objective space. ParetoTransport guides a flow-matching sampler to iteratively transport the empirical offline distribution toward the Pareto front, with Wasserstein matching to intermediate proxy distributions. This directly controls distributional displacement and mass allocation along the front. We establish a convergence result and demonstrate state-of-the-art performance on standard offline MOO benchmarks, extending recent evaluations beyond hypervolume to generational distance, inverted generational distance, and Wasserstein distance.

Stephanie Holly, Sepp Hochreiter, Werner Zellinger

LIT AI Lab and Institute for Machine Learning JKU Linz, Austria

Multi-Objective Optimization Pareto Optimization

Aug 5, 2026cs.LG

Cost-Aware Multi-Objective Bandits: Theory and Application to Budgeted LLM Configuration Evaluation

Large language model (LLM) configuration evaluation is challenging due to limited evaluation budgets, varying costs, and multiple competing objectives. In this paper, we formulate LLM configuration evaluation as a cost-aware multi-objective bandit problem, where each configuration evaluation incurs a configuration-dependent cost and yields a noisy vector-valued outcome. Under this framework, we study two fundamental problems: online configuration selection and Pareto configuration identification. For online configuration selection, we propose a hypervolume-based UCB algorithm that optimizes an optimistic hypervolume-per-cost index. We establish a budgeted regret bound of order

O\bigl(\sum_{i\ne i^\star}\frac{\log B}{Δ_i}\bigr)

, where

B

is the evaluation budget,

i^\star

is the optimal configuration in terms of hypervolume efficiency, and

Δ_i

is the corresponding efficiency gap of configuration

i

. This bound retains the logarithmic budget dependence of classical single-objective budgeted bandits. For fixed-budget Pareto identification, we develop a cost-aware empirical gap elimination algorithm and prove that its error probability is of order

O\bigl(\exp(-\frac{B}{H_{μ,c}})\bigr)

, where

H_{μ,c}

is a cost-aware Pareto identification complexity depending on configuration costs and Pareto classification gaps. This error probability decays exponentially with the evaluation budget and recovers the standard Pareto set identification guarantee when all configuration costs are identical. Experiments on LLM configuration evaluation tasks demonstrate that the proposed framework enables efficient online decision-making and accurate cost-aware Pareto identification under limited budgets.

Bo Xue, Zhi Hong, Jiayi Li +3

City University of Hong Kong · South China University of Technology · Zhejiang University

Multi-Armed Bandits LLM Evaluation Multi-Objective Optimization Pareto Optimization

Jul 28, 2026cs.LG

Top- $k$ Pareto Bandits: Hypervolume Regret for Multi-Objective Slate Selection

We consider a stochastic multi-objective bandit problem where, at each round, the agent selects a slate of

k

arms and observes their

d

-dimensional reward vectors under semi-bandit feedback. We do not aim at identifying a single optimal arm; instead, we consider the problem of maintaining a small set of actions that jointly approximate the Pareto frontier. We formalize this objective through the dominated hypervolume induced by the selected subset of arms, and define an

α

-approximate hypervolume regret with respect to the best size-

k

subset achievable in hindsight, where

α= 1 - 1/e

reflects the approximation guarantee of greedy maximization for monotone submodular functions. To address this problem, we introduce \textit{THV-UCB}, an optimistic algorithm that selects arms greedily based on optimistic estimates of their marginal hypervolume contributions. We establish a gap-free regret bound

\tilde{O}(d\sqrt{nkT})

that holds on every instance, together with a gap-dependent bound

\tilde{O}(nk^{2.5}/Δ_{\min})

that becomes polylogarithmic in

T

once the arms are sufficiently well separated. Our results provide theoretical support for using small subsets to approximate Pareto fronts in various multi-objective applications.

Nicolas Gutowski, Fabien Chhel, Alexandre Letard +1

Université d’Angers, LERIA, SFR MATHSTIC, Angers, France · ESEO, ERIS, Angers, France · AlphaEdge, Saint-Barthélémy-d’Anjou, France +1

Multi-Armed Bandits Combinatorial Multi-Armed Bandits Multi-Objective Optimization Pareto Optimization

Jul 25, 2026cs.GT

Fair Division with Strictly Increasing Valuations: A Tight Threshold for Two-Agent EF1 and PO

We study whether strictly positive marginal values restore the compatibility of envy-freeness up to one good (EF1) and Pareto optimality (PO) for indivisible goods. For two agents, we identify the exact threshold in the number of goods. Every instance with at most seven goods and strictly increasing valuations admits an allocation that is both EF1 and PO, without any submodularity assumption. In contrast, we construct an eight-good instance with normalized, integer-valued, strictly increasing, submodular valuations in which every EF1 allocation is strictly Pareto dominated. Thus, eight goods are necessary and sufficient for a two-agent counterexample. Finally, we strengthen the three-agent NP-hardness result of Chandramouleeswaran and Nimbhorkar (2026): deciding whether an EF1 and PO allocation exists remains NP-hard for normalized, integer-valued, monotone submodular valuations even when zero marginals are confined to eight fixed agent-good pairs, all involving a single agent.

Nicholas Teh

University of Oxford, UK

Social Choice Theory Fair Resource Allocation Pareto Optimization

Jul 13, 2026cs.DM

Representing the Non-dominated Set of Multi-objective Network Problems by Supported Non-dominated Points

In multi-objective combinatorial optimization, unsupported non-dominated points typically outnumber supported points and are often significantly more challenging to compute. Recent studies show that extreme supported non-dominated points provide high-quality representations of the non-dominated set for certain binary problems. We demonstrate that this observation does not generalize to capacitated network optimization problems: representation quality decreases with increasing arc capacities, whereas supported non-dominated points consistently provide high-quality representations with respect to several quality indicators. However, supported point sets may still be too large in practical applications, where only a small, fixed number of alternatives is typically desired. Selecting fixed-size representations from the non-dominated set requires its computationally expensive generation and thus diminishes the computational advantages that representations are intended to provide. We therefore suggest the (extreme) supported points as alternative candidate sets in subset selection problems. Our numerical results show that restricting the candidate set to supported non-dominated points yields fixed-size representations of nearly the same quality as those selected from the complete non-dominated set. Overall, supported non-dominated points serve both as high-quality representations and as reasonable candidate sets for subset selection.

David Könen, Lara Löhken, Michael Stiglmayr

University of Wuppertal, School of Mathematics and Natural Sciences, Optimization Group, Gaußstraße 20, 42103 Wuppertal, Germany · Corresponding author

Combinatorial Optimization Multi-Objective Optimization Pareto Optimization

Jun 24, 2026cs.LG

Deterministic Pareto-Optimal Policy Synthesis for Multi-Objective Reinforcement Learning

Real-world decision-making often requires balancing multiple conflicting objectives, a challenge that standard Reinforcement Learning (RL) frequently addresses by aggregating rewards into a single scalar signal. While effective for simple tasks, this approach often fails to capture the full spectrum of optimal trade-offs, known as the Pareto frontier. In this paper, we introduce a novel preference-conditioned Bellman operator, motivated from the Chebyshev scalarization, designed to compute deterministic Pareto-optimal policies for Multi-Objective Markov Decision Processes (MOMDPs). We prove that this operator satisfies an enveloping property, where the estimated value functions upper-bound the true Pareto frontier, and demonstrate that it monotonically converges to a coverage set of this frontier. Furthermore, we also show how to extract deterministic policies from these converged Q-estimates. This ensures the agent can recover a policy for any given preference, capturing the entire Pareto-optimal frontier while guaranteeing each synthesized policy remains approximately Pareto-optimal. Experimental results validate that our algorithm successfully recovers complex trade-offs, providing a solution for deterministic Pareto-optimal policy synthesis.

Aniruddha Joshi, Niklas Lauffer, Sanjit Seshia

University of California Berkeley, USA.

Multi-Objective Reinforcement Learning Multi-Objective Optimization Pareto Optimization

Jun 20, 2026cs.LG

Selective Ensemble Based on Preference-Directed Multi-Objective Bandits

Selective ensemble for modern machine learning systems requires choosing promising model candidates under limited evaluation budgets, while downstream tasks often specify only partial preferences over capabilities such as accuracy, robustness, and reasoning. This setting naturally gives rise to a sequential decision problem under partially specified linear preferences. We formalize it as preference-directed multi-objective bandits (PDMOB), where admissible trade-offs are represented by a polyhedral preference cone. Based on this formulation, we introduce Pareto

C

-optimality, which recovers standard Pareto optimality and single-weight scalarization as special cases. We then propose the preference-directed upper confidence bound (PrefUCB) algorithm, which maintains directional confidence intervals to guide exploration. We analyze both indicator-based and gap-weighted regret, and establish instance-dependent logarithmic bounds for both criteria, recovering the optimal logarithmic dependence on the horizon

T

in classical special cases. Experiments on large pre-trained model selective ensemble tasks and online asset allocation under institutional mandates validate the efficacy of our method.

Lanjihong Ma, Zhen-Yu Zhang, Masashi Sugiyama +1

Department of Computer Science, Zhejiang Gongshang University, China · Center for Advanced Intelligence Project, RIKEN, Japan · Graduate School of Frontier Sciences, The University of Tokyo, Japan +1

Multi-Armed Bandits Ensemble Learning UCB Algorithms Combinatorial Multi-Armed Bandits Pareto Optimization Preference Learning

Jun 19, 2026cs.GT

Simultaneously Efficient Allocation of Indivisible Items Across Multiple Dimensions

Many allocation problems are intrinsically multidimensional, since an item may contribute differently to several criteria, and optimizing a single aggregate objective can hide severe losses in other dimensions. We study how much efficiency can be guaranteed simultaneously when indivisible items have multiple attributes. To this end, we introduce the \emph{multidimensional efficient allocation} (MDEA) model, where each agent has an additive valuation in each dimension, and investigate simultaneous efficiency under utilitarian social welfare (USW) and egalitarian social welfare (ESW). Our results reveal a sharp worst-case frontier. For exact efficiency, maximizing the number of dimensions attaining the USW optimum admits a

c/\ell

-approximation for every fixed constant

c

, and this dependence on the number

\ell

of dimensions is essentially unavoidable; for ESW, even deciding whether two dimensions can be optimized simultaneously is NP-hard with binary valuations. For approximate simultaneous efficiency in every dimension, we identify a tight threshold of order

1/\ell

, showing that such guarantees always exist for both USW and ESW, while any asymptotically better dependence on

\ell

is impossible, even for binary valuations. Finally, we introduce three natural multidimensional Pareto notions and characterize both their relationships and their computational complexity.

Yasushi Kawase, Bodhayan Roy, Mohammad Azharuddin Sanpui

Chuo University, Tokyo, Japan · Indian Institute of Technology Kharagpur, Kharagpur, India

Combinatorial Optimization Approximation Algorithms Multi-Objective Optimization Pareto Optimization

Jun 17, 2026cs.LG

Interactive Pareto navigation for deep multi-task learning

In multi-task learning, handling an increasing number of objectives can quickly become challenging, both in terms of the computational resources and the decision maker's capacity to choose appropriate trade-offs. A widely used approach is thus to aggregate the individual losses in a single loss function by a weighted sum. This often fails to capture either the decision maker's preferences as a result of the shape of the Pareto front, or requires multiple adjustments and computations which becomes prohibitively expensive in deep learning applications. To address these issues, we introduce a novel framework, Preference Pareto Exploration (PPE), which enforces the decision maker's preferences while accounting for the geometry of the Pareto set in an interactive exploration process. PPE is based on a predictor-corrector method that performs predictor steps tangential to the manifold of Pareto-optimal solutions, following the decision maker's preference. The subsequent corrector step results in a new trade-off reflecting this preference. To avoid explicit Hessian computations when characterizing the tangent space of the manifold, we employ a Krylov subspace method that relies solely on matrix-vector products. These products can be efficiently obtained via automatic differentiation, ensuring both efficiency and robustness throughout the optimization process. The method's functionality and performance are demonstrated using both toy problems and examples from deep learning.

Augustina C. Amakor, Konstantin Sonntag, Sebastian Peitz

Department of Computer Science, TU Dortmund, Dortmund, Germany · Lamarr Institute for Machine Learning and Artificial Intelligence

Multi-Task Learning Pareto Optimization Preference Learning

Jun 17, 2026cs.LG

Bayesian Anytime Pareto Set Identification for Multi-Objective Multi-Armed Bandits

Identifying Pareto optimal solutions is critical to support multi-objective decision-making. We introduce the first anytime Multi-Objective Multi-Armed Bandit algorithm for the Pareto Set Identification problem, taking a Bayesian approach: Top-Two Pareto Front Thompson Sampling (TTPFTS). We benchmark TTPFTS against state-of-the-art fixed-budget Pareto Set Identification algorithms on synthetic environments. Next, we demonstrate its practical utility in a challenging multi-objective molecular discovery setting by efficiently exploring an ultra-large synthesis-on-demand molecular library. Furthermore, we introduce a novel uncertainty quantification metric that estimates our algorithm's confidence in the predicted Pareto set. We demonstrate that this metric effectively proxies true performance, yielding a robust methodology for monitoring learning progress in complex settings. Finally, we complement these empirical findings with a theoretical proof of the algorithm's asymptotic correctness.

Lennert Saerens, Bram Silue, Eleni Litsa +2

Artificial Intelligence Lab, Departement of Computer Science, Vrije Universiteit Brussel, Brussels, Belgium · imec, Leuven, Belgium

Thompson Sampling Multi-Armed Bandits Combinatorial Multi-Armed Bandits Pareto Optimization

Jun 15, 2026cs.CL

Towards Pareto-Optimal Tool-Integrated Agents with Pareto Ranking Policy Optimization

Recent advances in tool-integrated language agents have significantly improved their ability to solve complex reasoning tasks. However, existing alignment methods predominantly focus on maximizing task accuracy, while overlooking auxiliary objectives such as tool-use efficiency, which are essential for practical deployment. To address this gap, we introduce ParetoPO, a two-stage multi-objective optimization framework for aligning tool-using large language models (LLMs) under competing objectives. In the first stage, ParetoPO leverages hypervolume-guided dynamic scalarization to adapt reward weights based on global Pareto frontier progress. In the second stage, it replaces scalarized learning signals with Pareto-ranking-based advantage computation, promoting nondominated trajectories through dominance-aware credit assignment. This design enables fine-grained, action-level optimization across multiple conflicting objectives. Experimental results on mathematic reasoning and multi-hop QA tasks show that ParetoPO consistently discovers policies with superior accuracy-efficiency trade-offs compared to static and heuristic baselines.

Junyi Li, Xiaowei Qian, Yingyi Zhang +6

Department of Data Science, City University of Hong Kong, Hong Kong, China · Zhejiang University of Technology, Hangzhou, China · Huawei Technologies Noah’s Ark Lab, Hong Kong, China

Multi-Objective Reinforcement Learning Multi-Objective Language Model Alignment Pareto Optimization RL for Tool Use

Jun 13, 2026cs.LG

Diversity-Driven Offline Multi-Objective Optimization via Nested Pareto Set Learning

Multi-objective optimization (MOO) has emerged as a powerful approach to solving complex optimization problems involving multiple objectives. In many practical scenarios, function evaluations are unavailable or prohibitively expensive, necessitating optimization solely based on a fixed offline dataset. In this setting, known as offline MOO, the goal is to find out the Pareto set without access to the true objective functions. This setting suffers from the out-of-distribution (OOD) issue, where the surrogate model is not accurate for unseen designs. Due to the OOD issue, surrogate errors may cause the optimizer to select solutions that do not lie on the true Pareto front and are biased toward its extremes. To address this, this paper proposes Diversity-driven Offline Multi-Objective Optimization (DOMOO), which aims to find out a diverse and high-quality set of solutions. First, DOMOO incorporates an accumulative risk control module that estimates the potential risk of candidate solutions and alleviates the OOD issue between the training data and the generated solutions. In addition, a nested Pareto set learning (PSL) strategy is proposed to jointly learn preference and PSL parameters, then optimize them, enabling adaptation to diverse Pareto front geometries. To further enhance solution quality, we design a diversity-driven selection strategy that extracts a representative and well-distributed set of final solutions. To achieve this diversity-driven selection strategy, we propose

\text{IGD}_\text{offline}

, a tailored indicator for the offline setting that considers both diversity and convergence, and avoids the bias of hypervolume indicator. Extensive experiments on synthetic and real-world benchmarks show that DOMOO achieves the best average rank across tasks in both convergence and diversity among the compared methods.

Yiyi Zhu, Yaolin Wen, Xiang Xia +6

Shanghai Institute of AI for Education, and School of Computer Science and Technology, East China Normal University, Shanghai, China · Ant Group, Hangzhou, China · Shanghai Innovation Institute, Shanghai, China

Surrogate-Assisted Optimization Multi-Objective Optimization Pareto Optimization

Jun 6, 2026cs.GT

Post-AGI Economies: Superposition and the Second Fundamental Theorem of Welfare Economics

The classical Second Welfare Theorem decentralizes any Pareto efficient allocation through prices and transfers under convexity and regularity. In post AGI economies, autonomy rights, self-modification, identity continuity, and superposed preferences need not behave as commodities or define a stable welfare relation, so this reduction may fail even when a supporting hyperplane exists. We give an autonomy-qualified Second Welfare Theorem stating the joint conditions convexity, stable moral status, non-fungible rights, welfare selection, non manipulation, governed self modification, and verification under which an autonomy Pareto optimum remains certifiably decentralizable, distinguishing economic preference superposition, a hypothesis about context-indexed choice, from neural feature superposition.

Elija Perrier

Centre for Quantum Software & Information, University of Technology Sydney, Sydney, Australia

AI Agent Governance Pareto Optimization

May 20, 2026cs.LG

SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front

Scalarization is widely used in multi-objective optimization owing to its simplicity and scalability. In many applications, the goal is to generate solutions that represent diverse user preferences, ideally with uniform coverage of the Pareto front (PF). However, uniformly sampling scalarization weights usually induces non-uniform coverage of the PF. We explain this mismatch through a geometric analysis of the scalarization path. As the scalarization weight varies, the corresponding solutions trace the PF with a generally non-uniform traversal speed. This speed induces an arc-length cumulative distribution function (CDF); inverting this CDF map yields a principled rule for selecting weights that produce uniform PF coverage. Building on this insight, we propose SURF (Sampling Uniformly along the PaReto Front). For structured problems, including bi-objective bandits, we derive closed-form expressions for this CDF map and the resulting PF-aware weight sampling rule. For general problems, SURF alternates between CDF reconstruction and weight sampling. Theoretically, we show that under provable conditions, SURF converges linearly to an unavoidable finite-sampling floor. Empirically, experiments on bandits, multi-objective-gymnasium, and multi-objective LLM alignment demonstrate that SURF efficiently achieves more uniform PF coverage than baselines.

Liuyuan Jiang, Chentong Huang, Lisha Chen

Department of Electrical and Computer Engineering University of Rochester, Rochester, NY 14627

Adaptive Sampling Multi-Objective Optimization Pareto Optimization

May 19, 2026cs.AI

MOCHA: Multi-Objective Chebyshev Annealing for Agent Skill Optimization

LLM agents organize behavior through skills - structured natural-language specifications governing how an agent reasons, retrieves, and responds. Unlike monolithic prompts, skills are multi-field artifacts subject to hard platform constraints: description fields are truncated for routing, instruction bodies are compacted via progressive disclosure, and co-resident skills compete for limited context windows. These constraints make skill optimization inherently multi-objective: a skill must simultaneously maximize task performance and satisfy platform limits. Yet existing prompt optimizers either ignore these trade-offs or collapse them into a weighted sum, missing Pareto-optimal variants in non-convex objective regions. We introduce MOCHA (Multi-Objective Chebyshev Annealing), which replaces single-objective selection with Chebyshev scalarization - covering the full Pareto front, including non-convex regions - combined with exponential annealing that transitions from exploration to exploitation. In our experiments across six diverse agent skills - where all methods share the same multi-objective mutation operator and baselines receive identical per-objective textual feedback - existing optimizers fail to improve the seed skill on 4 of 6 tasks: 1000 rollouts yield zero progress. MOCHA breaks through on every task, achieving 7.5% relative improvement in mean correctness over the strongest baseline (up to 14.9% on FEVER and 10.4% on TheoremQA) while discovering twice as many more Pareto-optimal skill variants.

Md Mehrab Tanjim, Jayakumar Subramanian, Xiang Chen +6

Adobe Research

LLM Agent Skill Learning Multi-Objective Optimization Pareto Optimization

May 19, 2026cs.LG

A Two-Phase Adaptive Balanced Penalty Method for Controllable Pareto Front Learning under Split Feasibility Conditions

We address the open problem of training hypernetworks for Controllable Pareto Front Learning (CPFL) under split feasibility conditions with rigorous theoretical guarantees. We reformulate the constrained Pareto problem as a Bi-Level Scalarized Split Problem (BSSP) and propose the Adaptive Balanced Penalty (ABP) algorithm, whose three gradient components -- optimality, set feasibility, and image feasibility -- are blended through an adaptive indicator driven by a computable lower bound. Using a novel convex surrogate technique, we prove full-sequence convergence under standard convexity and Robbins-Monro step-size assumptions. The ABP penalty structure is then translated into a two-phase, feasibility-first training strategy for Hyper-MLP and HyperTrans architectures (ABP-HyperNet). To evaluate constrained CPFL, we introduce the Expected Feasible Hypervolume (EFHV), which jointly captures solution quality and constraint satisfaction. Experiments on five multi-objective benchmarks validate the ABP solver against ground truth, while three multi-task learning datasets demonstrate that ABP-HyperNet achieves up to 2.3x higher EFHV than unconstrained baselines by raising feasibility from 36-49% to 87-100%.

Nguyen Viet Hoang, Dung D. Le, Tran Ngoc Thang

Hypernetworks Multi-Objective Optimization Pareto Optimization

May 13, 2026math.OC

Nonsmooth Set-Gradient Ascent to the Pareto Front via Layered Hypervolume and Magnitude Indicators

A nonsmooth set-gradient ascent method is developed for moving finite approximation sets toward the Pareto front in multiobjective optimization. The method optimizes layered set indicators: a base indicator is evaluated on successive nondomination layers, and the layer values are combined with rapidly decreasing weights. This gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front. Two base indicators are treated: the hypervolume indicator and the magnitude indicator of the dominated set, whose expansion over coordinate projections contains extent, projected-area, and volume terms. The scalar objectives are nonsmooth because nondomination layers change combinatorially and the active orthogonal-union geometry changes piecewise. On fixed strata, where layer assignments and active geometry remain unchanged, the indicators are piecewise smooth and chamberwise continuous. For the magnitude indicator, an exact gradient formula is derived as a linear combination of hypervolume gradients of projected shadow sets. Thus, for fixed objective dimension, magnitude gradients have the same asymptotic time complexity as hypervolume gradients. Lexicographic layer aggregation is related to a unary infinitesimal encoding. For finite-

ε

surrogates, the main nonsmoothness mechanisms are isolated and chamberwise Lipschitz continuity on bounded sets is proved; a two-point counterexample shows that hard-layer scalarization is not globally continuous across layer switches. The theory motivates a projected finite-difference implementation with repulsion and recovery from stagnation. Numerical examples and reproducible code cover two- and three-objective settings, including objective-space tests, curved fronts, a supersphere benchmark, and traces comparing layered magnitude and hypervolume ascent.

Michael T. M. Emmerich

Faculty of Information Technology, University of Jyväskylä, Finland

Multi-Objective Optimization Pareto Optimization Nonsmooth Convex Optimization

May 12, 2026cs.AI

Large Language Models as Amortized Pareto-Front Generators for Constrained Bi-Objective Convex Optimization

Generating feasible Pareto fronts for constrained bi-objective continuous optimization is central to multi-criteria decision-making. Existing methods usually rely on iterative scalarization, evolutionary search, or problem-specific solvers, requiring repeated optimization for each instance. We introduce DIPS, an end-to-end framework that fine-tunes large language models as amortized Pareto-front generators for constrained bi-objective convex optimization. Given a textual problem description, DIPS directly outputs an ordered set of feasible continuous decision vectors approximating the Pareto front. To make continuous optimization compatible with autoregressive language modeling, DIPS combines a compact discretization scheme, Numerically Grounded Token Initialization for new numerical tokens, and Three-Phase Curriculum Optimization, which progressively aligns structural validity, feasibility, and Pareto-front quality. Across five families of constrained bi-objective convex problems, a fine-tuned 7B-parameter model achieves normalized hypervolume ratios of 95.29% to 98.18% relative to reference fronts. With vLLM-accelerated inference, DIPS solves one instance in as little as 0.16 seconds and outperforms general-purpose and reasoning LLM baselines under the evaluated setting. These results suggest that LLMs can serve as effective amortized generators for continuous Pareto-front approximation.

Peipei Xu, SiYuan Ma, Yaohua Liu +4

University of Shanghai for Science and Technology · The Hong Kong University of Science and Technology

Large Language Model-Guided Optimization Constrained Optimization Multi-Objective Optimization Pareto Optimization

May 11, 2026cs.LG

Fairness vs Performance: Characterizing the Pareto Frontier of Algorithmic Decision Systems

Designing fair algorithmic decision systems requires balancing model performance with fairness toward affected individuals: More fairness might require sacrificing some performance and vice versa, yet the space of possible trade-offs is still poorly understood. We investigate fairness in binary prediction-based decision problems by conceptualizing decision making as a multi-objective optimization problem that simultaneously considers decision-maker utility and group fairness. We investigate the set of Pareto-optimal decision rules for arbitrary utility functions for decision maker, arbitrary population distributions, and a wide range of group fairness metrics. We find that the Pareto frontier consists of deterministic, group-specific threshold rules applied to individuals' success probability. This complements existing optimality theorems from literature which, for specific fairness constraints, posit lower-bound threshold rules only. However we also show that, depending on the used fairness metric, the Pareto frontier may include upper-bound threshold rules, thus preferring individuals with lower success probabilities. We show that the location of the Pareto frontier depends only on population characteristics, utility functions and fairness score, but not on the technical design of the algorithm - our findings hold for pre-, in-, and post-processing approaches alike. Our results generalize existing optimality theorems for fairness-constrained classification and extend them to generalized fairness metrics and fairness principles, and to partial fairness regimes. This paper connects formal fairness research with legal and ethical requirements to search for less discriminatory alternatives, offering a principled foundation for evaluating and comparing algorithmic decision systems.

Mieke Wilms, Christoph Heitz

Zurich University of Applied Sciences / University of Zurich, Switzerland · Zurich University of Applied Sciences, Switzerland

Algorithmic Fairness Group Fairness Multi-Objective Optimization Pareto Optimization

May 8, 2026cs.AI

From Feasible to Practical: Pareto-Optimal Synthesis Planning

Current computer-aided synthesis planning (CASP) methods often treat retrosynthesis as solved once a single feasible route is identified, focusing primarily on convergence or shortest-path metrics. This view is misaligned with real-world practice, where chemists must balance competing objectives such as cost, sustainability, toxicity, and overall yield. To address this, we formulate synthesis planning as a multi-objective search problem and introduce MORetro*, an algorithm that generates a Pareto front of synthesis routes to explicitly capture trade-offs among user-defined criteria. MORetro* uses weighted scalarization and BO-informed sampling to efficiently navigate the combinatorial search space and prioritize promising trade-offs. Building on multi-objective A*-search, we provide optimality guarantees showing that, for a fixed single-step model, MORetro* recovers the true Pareto front under admissibility. Across multiple retrosynthesis benchmarks, MORetro* produces diverse, high-quality Pareto fronts, uncovering solutions overlooked by single-objective approaches and better aligning CASP outputs with industrial decision-making.

Friedrich Hastedt, Dongda Zhang, Antonio del Rio Chanona

Department of Chemical Engineering, Imperial College London, UK · Department of Chemical Engineering, University of Manchester, UK

Multi-Objective Optimization Pareto Optimization

May 3, 2026cs.LG

CoAction: Cross-task Correlation-aware Pareto Set Learning

Pareto set learning (PSL) is an emerging paradigm in multi-objective optimization that trains neural networks to map preference vectors to Pareto optimal solutions. However, existing PSL methods primarily focus on solving a single multi-objective optimization problem at a time. This limitation not only increases computational costs in multi-objective multitask optimization scenarios by requiring a separate model for each task, but also fails to exploit the inter-task correlations across tasks. To address this, we propose a Cross-tAsk correlation-aware Pareto Set Learning (CoAction) framework, which leverages task-aware transformer to handle multiple tasks simultaneously. Specifically, by assigning task-specific embedding vectors to individual tasks, the model effectively distinguishes between tasks while facilitating knowledge sharing among them. We utilize a Transformer encoder as the backbone architecture to leverage its self-attention mechanism for capturing complex task dependencies. The proposed approach is evaluated on comprehensive multitask test suites covering both benchmark problems and real-world applications, demonstrating effectiveness and competitive performance in Hypervolume, Range, and Sparsity.

Xinyue Chen, Yingxuan Liang, Yiqin Huang +3

Guangdong University of Technology, Guangzhou, China · Xiamen University, Xiamen, China

Multi-Task Learning Multi-Task Optimization Multi-Objective Optimization Pareto Optimization

Apr 27, 2026cs.CV

POCA: Pareto-Optimal Curriculum Alignment for Visual Text Generation

Current visual text generation models struggle with the trade-off between text accuracy and overall image coherence. We find that achieving high text accuracy can reduce aesthetic quality and instruction-following capability. Although reinforcement learning approaches can alleviate the problem through aligning with multiple rewards, they are often unstable for text generation, as existing approaches normally optimize multiple rewards in a weighted-sum way. In addition, it is difficult to balance the weight of each reward. Moreover, reinforcement learning requires a set of training instructions. A large number of prompts require more training time and computing resources, while a small set leads to poor performance. Hence, how to select the prompts for efficient training is an unsolved problem. In this study, we propose Pareto-Optimal Curriculum Alignment (POCA), a framework that addresses this issue as a multi-objective problem by: 1) identifying the Pareto-optimal set to avoid simple scalarization and 2) designing an adaptive curriculum alignment strategy to manage a learning sequence of a multi-reward dataset using automatic difficulty assessment, which is crucial for optimal convergence as RL methods explore in a limited data environment. In synergy, POCA finds the Pareto-optimal set in a unified reward space, which eliminates inconsistent signals to find the best trade-off solution from different rewards under an easy-to-hard optimization landscape. The experimental results show that POCA significantly improves all metrics such as CLIP, HPS scores and sentence accuracy.

Yaohou Fan, Qingzhong Wang, Yongsong Huang +3

Tohoku University · Amazon Web Services

Multi-Objective Reinforcement Learning RL for Image Generation Curriculum Learning Pareto Optimization

Apr 20, 2026math.OC

The Magnitude of Dominated Sets: A Pareto Compliant Indicator Grounded in Metric Geometry

We investigate \emph{magnitude} as a new unary and strictly Pareto-compliant quality indicator for finite approximation sets to the Pareto front in multiobjective optimization. Magnitude originates in enriched category theory and metric geometry, where it is a notion of size or point content for compact metric spaces and a generalization of cardinality. For dominated regions in the

\ell_1

box setting, magnitude is close to hypervolume but not identical: it contains the top-dimensional hypervolume term together with positive lower-dimensional projection and boundary contributions. This paper gives a first theoretical study of magnitude as an indicator. We consider multiobjective maximization with a common anchor point. For dominated sets generated by finite approximation sets, we derive an all-dimensional projection formula, prove weak and strict set monotonicity on finite unions of anchored boxes, and thereby obtain weak and strict Pareto compliance. Unlike hypervolume, magnitude assigns positive value to boundary points sharing one or more coordinates with the anchor point, even when their top-dimensional hypervolume contribution vanishes. We then formulate projected set-gradient methods and compare hypervolume and magnitude on biobjective and three-dimensional simplex examples. Numerically, magnitude favors boundary-including populations and, for suitable cardinalities, complete Das--Dennis grids, whereas hypervolume prefers more interior-filling configurations. Computationally, magnitude reduces to hypervolume on coordinate projections; for fixed dimension this yields the same asymptotic complexity up to a factor

2^d-1

, and in dimensions two and three

Θ(n\log n)

time. These results identify magnitude as a mathematically natural and computationally viable alternative to hypervolume for finite Pareto front approximations.

Michael T. M. Emmerich

Faculty of Information Technology, University of Jyväskylä, Finland

Multi-Objective Optimization Pareto Optimization