Multi-Objective Optimization

Motor imagery (MI) classification using electroencephalography (EEG) signals is essential for advancing brain-computer interfaces (BCIs). Traditional EEG channel selection methods often face limitations, such as dependency on single-objective criteria and susceptibility to local optima. To address these challenges, this work proposes a multi-objective optimisation framework that employs non-dominated sorting genetic algorithm, multiple-objective particle swarm optimisation, and a multi-objective evolutionary algorithm based on decomposition. Our approach effectively balances spatial relevance, using a Gaussian kernel, and functional discriminability, which assesses intratrial task-related desynchronisation, thereby improving performance. We evaluated this framework on four EEG datasets: Physionet, OpenBMI, HighGamma, and BCIIV-2A. The proposed approach successfully identifies compact, relevant channel subsets concentrated around sensorimotor cortex regions linked to MI activity, addressing the prevalent challenges of dimensionality and complexity inherent to traditional techniques. Furthermore, the framework achieved classification performance of 87%, 71%, 75%, and 65% on the Physionet, OpenBMI, HighGamma, and BCIIV-2A datasets, respectively. By outperforming existing single-objective and accuracy-based methods, and those relying on fixed subsets, these findings demonstrate that this new multi-objective optimisation framework can enhance MI-based BCI performance while facilitating compact channel configurations with reduced computational complexity, making them better suited for wearable, portable, and real-time BCI applications.

Applying the concepts and formalism from Evolutionary Game Theory to the data regime, the fundamental paradigms of Evolutionary Data Theory are introduced. It is shown that essential definitions and results such as replicator equations, evolutionary strategies, the Bishop-Cannings theorem and the analogy to Lotka-Volterra systems can be mapped to the data interpretation. Understanding data in matrix form as evolutionary entities, input data is mapped to genes and organisms. Steered by genetic fitness and two evolutionary strategies, Dominant-Balanced and Altruistic-Selfish, data records and features conduct an evolutionary game. It is shown that this evolutionary interpretation remains universally meaningful, by proving convergence to a unique rest point, where all data features persist in the population. A basic example of multi-objective optimization is shown as well as a related distribution problem and machine learning applications.

Federated edge learning (FEEL) has recently emerged as a promising paradigm for achieving edge intelligence (EI) via enabling collaborative model training across edge devices while protecting data privacy. In this paper, we put forth an online optimization framework that jointly manages federated training and inference on resource-constrained edge devices. We introduce a tandem-queue-inspired conversion mechanism that bridges inference requests and training data, and further incorporate both data and model freshness into the accuracy formulation to capture temporal dynamics in real-world environments. To maximize inference accuracy while minimizing latency and energy consumption, the mode selections, communication, and computation resource allocations of edge devices are jointly optimized. We formulate this optimization as a multi-objective optimization problem, which is NP-hard and further complicated by the online setting. To address these challenges, we transform the problem into a multi-objective Markov decision process (MOMDP) and develop a \underline{c}onstrained \underline{m}ulti-\underline{o}bjective \underline{p}roximal \underline{p}olicy \underline{o}ptimization (C-MOPPO) algorithm. Specifically, C-MOPPO first learns a set of policies with different preferences across three objectives, then leverages constrained policy optimization to enrich the Pareto front and obtain high-quality, dense solutions. Extensive experiments demonstrate that C-MOPPO achieves well-balanced trade-offs among objectives and significantly outperforms baselines under various system configurations.

Dynamic multi-objective optimization with a changing number of objectives has recently attracted increasing attention due to its relevance to real-world problems whose evaluation criteria may evolve over time. However, existing benchmark test suites for this problem setting suffer from a fundamental limitation: when the number of objectives changes, the objective functions themselves also change implicitly. This makes it difficult to isolate and evaluate an algorithm's capability to handle dynamics in the number of objectives alone. In this paper, we analyze this issue in detail and show that several theoretical properties claimed in prior studies rely on an assumption that is violated by commonly used test suites. To address this problem, we propose a scalable benchmark test suite in which the objective functions are fixed throughout the optimization process, while the number of active objectives changes over time. Our benchmark is constructed by defining a maximum-objective problem and dynamically selecting subsets of objectives. To avoid degeneracy issues in classical DTLZ and WFG problems, we adopt Minus-DTLZ and Minus-WFG formulations, in which all objectives are mutually conflicting. Extensive benchmark studies using representative algorithms from the literature demonstrate the usefulness and flexibility of the proposed test suite.

Kinodynamic planning for car-like vehicles on uneven terrain requires simultaneously optimizing competing objectives such as path efficiency and pose stability. This work presents an adaptive epsilon-constraint method integrated into a Model Predictive Control (MPC) framework, where the epsilon bounds are dynamically adjusted based on terrain descriptors to explore the Pareto front in real time. To capture vehicle-terrain dynamics, we develop a semi-parametric model combining analytical vehicle dynamics with a Sparse Gaussian Process (SGP) trained on the same terrain descriptors. The proposed epsilon-MPC is evaluated against MPPI and GAKD baselines, achieving a 94% navigation success rate while reducing maximum orientation deviation by 24% and improving multi-objective trade-off quality by 23%.

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.

Existing neural combinatorial optimization solvers frame solution search as imitation of optimal decisions, inherently limiting their utility to single-objective minimization and static constraints. We propose GOAL, a conditioned diffusion solver over relational graph representations that enables controllable decision generations by conditioning on human-specified objectives. We introduce a heterogeneous graph encoding in which distinct edge types, corresponding to different classes of constraints, define the message passing structure of the graph neural network, which allows information to propagate selectively according to the ontology of each constraint. GOAL is instantiated and evaluated on three canonical scheduling benchmarks of various constraint complexity: the Flow Shop Problem (FSP), the Job Shop Scheduling Problem (JSP), and the Flexible Job Shop Scheduling Problem (FJSP). Generalization is demonstrated across structurally distinct constraint regimes and problem types without architectural modification. On all three benchmarks, GOAL achieves 100% solution feasibility and near-zero MAPE (below 0.20%) on multiple objectives for problem sizes up to 20 jobs and 60 operations, outperforming NSGA-II and MOEA/D in both solution quality and inference speed by up to 25x.

We present Lean Refactor, a plug-and-play retrieval-augmented agentic framework for multi-objective, controllable, and version-robust refactoring of Lean proofs. LLM-generated proofs are notoriously correct-but-verbose and brittle across library versions, yet existing refactoring works overlook three practical challenges: 1) Lean refactoring is natively multi-objective (proof length, compilation cost, and version compatibility are often in tension); 2) Lean repositories have fragile compatibility, whereas LLM releases are unaware of Lean/Mathlib versions; 3) Training-based pipelines require repeated fine-tuning with each new LLM release, scaling neither with model churn nor with Lean's release cycle. Lean Refactor steers a frozen agentic LLM with retrievals from a curated database of multi-objective refactoring strategies, each densely annotated with metadata such as supported Lean/Mathlib versions and expected compilation-cost reduction. Experiments show over $70\%$ token-level compression on competition benchmarks, over $20\%$ on research repositories, and up to $60\%$ compilation-time reduction, outperforming prior work and Claude Code. Version-filtered retrieval further improves compression on the target Lean version, and refactored miniF2F proofs exhibit stronger zero-shot version transfer to future Lean releases than their unrefactored counterparts.

Physical adversarial attacks often overfit single surrogate models and optimization objectives. While ensemble attacks can mitigate this, existing methods struggle with severe gradient conflicts within restricted physical texture spaces, significantly degrading cross-model transferability. To bridge this gap, this paper proposes a Joint Multi-Objective and Multi-Model Optimization Framework (JMOF) that leverages quantitative similarity analysis to select the optimal surrogate model ensemble. Within JMOF, a dual-level mechanism jointly suppresses prediction outputs and flattens intermediate feature distributions, balancing attack efficiency with deep generalization. Additionally, an Orthogonal Gradient Alignment (OGA) strategy resolves cross-model gradient conflicts, transforming mutually repulsive gradients into synergistic optimization directions. Extensive simulated and real-world experiments demonstrate that JMOF outperforms state-of-the-art baselines against diverse black-box detectors. Crucially, JMOF exhibits substantial cross-vision-task generalization, generating attacks capable of simultaneously deceiving object detection and semantic segmentation or monocular depth estimation models. This research advances the generalization limits of physical adversarial attacks, providing a robust framework for evaluating visual AI vulnerabilities in real-world deployments.

Multi-party multi-objective optimization problems (MPMOPs) require consensus among autonomous decision makers and therefore differ from flattened many-objective formulations. Existing runtime theory for multi-objective evolutionary algorithms is largely tailored to single-party Pareto-front approximation and does not directly explain common-solution search in MPMOPs. We investigate cross-party recombination in two representative settings. On MP-JCG, a pseudo-Boolean benchmark with an explicit gap region, we prove that a payoff-guided mutation baseline faces a gap-crossing bottleneck requiring $Θ(n^2)$ expected fitness evaluations. In contrast, an analytical CPR-NSGA-II variant discovers both common Pareto-optimal solutions in $O(n\log n)$ expected evaluations by directly assembling complementary prefix and suffix templates distributed across party populations. Comparing this with the flattened four-objective formulation F-JCG, our full-front coverage analysis illustrates the additional coverage burden introduced by flattening. For BPBOMST, the bi-party, two-objective-per-party specialization of the multi-party multi-objective minimum spanning tree problem, we develop a layered support-cover analysis. For each common Pareto objective vector, the symmetric average projection induces an auxiliary bi-objective MST instance, and suitable support representatives yield a $2λ$-common approximation cover with $λ\in[1,2]$. We further derive an instance-parameterized expected runtime bound for a representative-pool CPR-NSGA-II variant using edge-union recombination and uniform repair. This bound separates the effects of local auxiliary-front filling, cross-party recombination shortcuts, and edge-union repair ambiguity.

Despite recent progress in constructing generalizable parallel algorithm portfolios (PAPs), no general-purpose approach is yet available for multi-objective binary optimization problems (MOBOPs). To fill this gap, this paper proposes domain-agnostic co-evolution of parameterized search for multi-objective binary optimization~(DACMO), which features two technical innovations. First, we propose a neural instance representation architecture that decouples domain-invariant and instance-specific features, enabling class-consistent instance generation across varying dimensions without problem-specific instance generators. Second, we introduce LLM-based automatic search operator generation into PAP construction, extending the search space from parameter tuning of predefined templates to operator-level algorithm design. We evaluate DACMO on four representative MOBOP classes to demonstrate its effectiveness as a general-purpose PAP construction method: the multi-objective match max problem~(MMMP), the multi-objective knapsack problem~(MKP), the multi-objective contamination control problem (MCCP), and the multi-objective complementary influence maximization problem~(MCIMP). Experimental results show that DACMO can be directly applied to all four problem classes without modification, outperforms PAPs built from classic MOEA templates, and achieves performance comparable to a privileged state-of-the-art baseline that relies on manually designed problem-specific instance generators, while outperforming it on two of the four evaluated problem classes.

Problems defined on binary decision spaces have been intensively studied in the theory of multi-objective evolutionary algorithms (MOEAs). In contrast, no mathematical runtime analyses exist so far for MOEAs dealing with decision variables that take a finite number $r > 2$ of values, despite the prevalence of such problems in practice. In this work, we begin to fill this research gap. We analyze how the classic SEMO algorithm with unit-strength local mutation computes the Pareto front of an $r$-valued counterpart of the classic \oneminmax benchmark. For the expected number of function evaluations until the Pareto front is covered by the population of this MOEA, we prove an upper bound of $O(n^2 r^2 \log n)$ and a near-tight lower bound of $Ω(n^2 r (r + \log n))$. We can close the small remaining gap between these two bounds by considering a variant of the algorithm that accepts only strictly better solutions; for this variant, we show an upper bound of $O(n^2 r (r + \log n))$, matching our lower bound (which also holds for this variant). Our results suggest that classic MOEAs encounter no significant additional difficulties when dealing with multi-valued decision variables. However, significantly more advanced tools may be required to obtain tight bounds for algorithms with more complex population dynamics.

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.

Large-scale multi-objective optimization problems (LSMOPs) remain challenging due to the high-dimensional decision spaces, complex variable interactions, and limited function evaluation budgets, which make it difficult to balance the convergence, diversity, and stability. Existing two-archive evolutionary algorithms can alleviate the conflict between convergence and diversity, but they often underuse archive reliability and problem-structure information, leading to inefficient search, incomplete front coverage, and late-stage archive drift. To address these issues, this paper proposes TRUST-TAEA, a trustworthiness-guided two-archive evolutionary algorithm. Archive trustworthiness is defined by integrating evolutionary progress with convergence-archive maturity, and is used to coordinate variable-grouping sparse search, anchor-probing compensatory search, and archive stabilization. TRUST-TAEA is evaluated on the LSMOP benchmark suite with 500--5000 decision variables and 2, 3-objectives. Experimental results show that TRUST-TAEA achieves superior and highly competitive performance in terms of convergence, diversity, and stability. A three-objective day-ahead scheduling case of a grid-connected microgrid further demonstrates its practical applicability, where TRUST-TAEA obtains the best IGD$^+$ value and generates a feasible dispatch strategy balancing cost, emissions, and grid-power fluctuation.

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.

In recent years, a theoretical understanding has rapidly advanced regarding how popular multi-objective evolutionary algorithms (MOEAs) can optimize many-objective problems. However, the benefits of using crossover in many-objective optimization are theoretically not understood, except for specifically designed benchmark functions tuned to particular crossover operators, and still lag significantly behind its practical use. In this paper, we build upon this line of research and present a theoretical runtime analysis of the widely used NSGA-III algorithm on the classical $m$-objective $m$-OneJumpZeroJump function ($m$-OJZJ for short). Our results demonstrate that NSGA-III with crossover optimizes $m$-OJZJ asymptotically faster than NSGA-III without crossover for any number $m$ of objectives for huge parameter regimes. We complement our analysis by providing a lower runtime bound on $4$-OJZJ when crossover is turned off.

Existing Meta-Black-Box Optimization (MetaBBO) methods focus on how to search when controlling optimizers, but largely overlook where to search. We propose MetaSG-SAEA, a bi-level MetaBBO framework for expensive constrained multi-objective optimization problems (ECMOPs), in which a meta-policy provides search guidance to the low-level Surrogate-Assisted Evolutionary Algorithm (SAEA). To achieve this, we introduce Max-Min Constraint-Calibrated Inequality (MM-CCI), a compact, problem-agnostic region abstraction that maps heterogeneous constraint evaluations to an ordered scalar level; we further provide a theoretical analysis of its fundamental properties. Building on this region abstraction, we adopt diffusion-based population initialization to translate the meta-policy's region-level guidance into solution-level priors for the SAEA. To make MetaSG-SAEA scalable, we construct an attention-based state representation across varying problem dimensions, population sizes, and numbers of objectives and constraints. Experimental results demonstrate that MetaSG-SAEA outperforms state-of-the-art baselines across diverse benchmarks and exhibits the ability to generalize across problem distributions.

Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial conditions must be optimized simultaneously. To address this difficulty, existing approaches often construct update directions by explicitly enforcing particular desirable properties, such as scale robustness and simultaneous descent. While effective in many cases, such property-by-property designs can make it unclear which conditions are essential, what geometric principle determines the selected update direction, and how different methods are structurally related. In this work, we formulate update-direction selection for PINN training as a Chebyshev-center problem in the dual cone. The proposed formulation selects a normalized direction that maximizes the minimum distance to the cone facets. The resulting formulation admits an efficient dual problem in a much lower-dimensional space and yields a convergence guarantee in the nonconvex setting. It also recovers the key desirable properties targeted by existing approaches without imposing them separately; rather, they follow from the single geometric criterion underlying the formulation. This makes the selected direction interpretable through a single geometric rule and provides a unified basis for systematically comparing related direction-selection methods. Experiments on several PINN benchmarks further demonstrate strong empirical performance of the proposed method.

Multi-object tracking (MOT) is a fundamental task in computer vision that requires continuously tracking multiple targets while maintaining consistent identities across frames. However, most existing approaches primarily rely on instance-level object features for trajectory association, which often leads to degraded performance under challenging conditions such as object deformation, nonlinear motion, and occlusion. In this work, we propose SAMOFT, a robust tracker that leverages pixel-level cues to improve robustness under complex motion scenarios. Specifically, we introduce a Pixel Motion Matching (PMM) module that integrates the Segment Anything Model (SAM) with dense optical flow to refine Kalman filter-based motion prediction using instantaneous foreground pixel motion. To further enhance robustness under unreliable detections, we design a Centroid Distance Matching (CDM) module that performs flexible mask-based centroid matching for low-confidence or partially occluded observations. Moreover, a Distribution-Based Correction (DBC) module models long-tailed motion patterns in a training-free manner using historical optical flow statistics and dynamically corrects trajectory states online. We also incorporate a Cluster-Aware ReID (CA-ReID) strategy to improve the stability and discriminative power of trajectory appearance features. Extensive experiments on the DanceTrack and MOTChallenge benchmarks demonstrate that SAMOFT consistently improves baseline trackers and achieves competitive performance compared with recent state-of-the-art methods, validating the effectiveness of leveraging pixel-level cues for robust multi-object tracking.

Preliminary low-thrust spacecraft mission design is a global search problem characterized by a complex solution landscape, multiple objectives, and numerous local minima. During this phase, mission parameters are often not yet fully defined, requiring new solutions to be generated at a high cadence across varying parameter values. When combined with the indirect approach to optimal control, diffusion models can accelerate this search by learning distributions that represent high-quality initial costates. However, generating training data remains expensive, and opportunities exist to better exploit past data. We propose a transfer-learning framework that combines homotopy in a mission parameter with Markov chain Monte Carlo (MCMC) to generate training data more efficiently. The approach reformulates a multiobjective optimization problem as sampling from an unnormalized target distribution in costate space. We compare three MCMC algorithms on a planar multi-revolution transfer in the circular restricted three-body problem, with homotopy in the system mass parameter. The results show that gradient-based MCMC variants achieve the best trade-off between sample quality and computational cost. For the test transfer, the proposed framework generates 40 % more feasible solutions and achieves a higher-quality Pareto front than a state-of-the-art indirect approach based on adjoint control transformations and gradient-based optimization. Finally, the MCMC-generated samples are used to fine-tune a diffusion model conditioned on the mass parameter, enabling it to learn a global representation of the underlying solution distribution and efficiently generate new solutions. These findings establish the transfer-learning framework as a practical method for efficiently solving indirect trajectory optimization problems with varying parameters.

In recent years, bilevel optimization (BLO) has attracted significant attention for its broad applications in machine learning. However, most existing works on BLO remain confined to the single-task setting and rely on the lower-level strong convexity assumption, which significantly restricts their applicability to modern machine learning problems of growing complexity. In this paper, we make the first attempt to extend BLO to the multi-task setting under a relaxed lower-level general convexity (LLGC) assumption. To this end, we reformulate the multi-task bilevel learning (MTBL) problem with LLGC into an equality constrained multi-objective optimization (ECMO) problem. However, ECMO itself is a new problem that has not yet been studied in the literature. To address this gap, we first establish a new Karush-Kuhn-Tucker (KKT)-based Pareto stationarity as the convergence criterion for ECMO algorithm design. Based on this foundation, we propose a weighted Chebyshev (WC)-penalty algorithm that achieves a finite-time convergence rate of O(ST^{-\frac{1}{2}) to KKT-based Pareto stationarity in both deterministic and stochastic settings, where $S$ denotes the number of objectives, and $T$ is the total iterations. Moreover, by varying the preference vector over the $S$-dimensional simplex, our WC-penalty method systematically explores the Pareto front. Finally, solutions to the ECMO problem translate directly into solutions for the original MTBL problem, thereby closing the loop between these two foundational optimization frameworks.

Conversational recommender systems (CRSs) are a core component of next-generation intelligent recommender systems because they enable users to actively elicit preferences, clarify intentions, and adapt recommendations in real time. However, there are two key obstacles in the CRS domain: evaluation and access to training data. Evaluating CRSs through real human studies is more critical than for traditional recommender systems, yet such studies are both costly and time-consuming. Moreover, CRS interaction data are often difficult to obtain for model training due to privacy concerns. Large language model (LLM)-based user simulators have shown promise in addressing both challenges by generating synthetic user interactions for evaluation and training. However, existing approaches suffer from systematic positive bias, data leakage, and limited behavioral diversity, and they rely on brittle manual prompt engineering that requires extensive domain expertise. In this paper, we propose a framework to automatically optimize prompts for LLM-based user simulators in CRSs, simultaneously mitigating these issues. Experimental results demonstrate that the proposed framework achieves improved behavioral alignment with human interaction patterns compared to baseline methods across diverse prompt settings.

Many real-world optimization problems consist of multiple tightly coupled subproblems whose solutions must be coordinated to achieve high overall performance. However, existing large language model driven automated heuristic design approaches are limited to single-problem settings. In this paper, we propose CoupleEvo. CoupleEvo proposes three evolutionary coordination strategies to evolve heuristics for coupled optimization problems: the sequential strategy evolves heuristics for one subproblem after the other; the iterative strategy alternates the evolution of heuristics for different subproblems over successive generations; and the integrated strategy evolves heuristics for all problems simultaneously. The approach is evaluated on two representative coupled optimization problems. Experimental results show that decomposition-based strategies (sequential and iterative) provide more stable convergence and higher solution quality, while the integrated evolution strategy suffers from increased search complexity and variability. These findings highlight the importance of coordinating evolutionary search across interdependent subproblems and demonstrate the potential of LLM-driven heuristic design for complex coupled optimization problems. The code is available: https://github.com/tb-git-kit-research/CoupleEvo.

Multi-objective optimization (MOO) has received growing attention in applications that require learning under multiple criteria. However, the existing MOO formulations do not explicitly account for distributional shifts in the data. We introduce distributionally robust multi-objective optimization (DR-MOO), which minimizes multiple objectives under their respective worst-case distributions. We propose Pareto-type solution concepts for DR-MOO and develop multi-gradient descent algorithms (MGDA) with provable guarantees. Leveraging a Lagrangian dual reformulation, we first design a double-loop MGDA that uses an inner loop to estimate dual variables and achieves a total sample complexity $\mathcal{O}(ε^{-12})$ for reaching an $ε$-Pareto-stationary point. To further improve efficiency, we incorporate gradient clipping to handle generalized-smooth and biased gradient estimates, removing the need for double sampling. This yields a single-loop double-clip MGDA with substantially improved sample complexity $\mathcal{O}(ε^{-4})$. Our theory applies to the nonconvex setting and does not require bounded objectives or gradients. Experiments demonstrate that our methods are competitive with state-of-the-art MGDA baselines.

Interactive multi-objective optimization systems face a budget allocation dilemma: one can spend resources on expensive objective evaluations or on eliciting decision-maker preferences that identify the relevant region of the Pareto set. Moreover, preference elicitation itself spans modalities with different information content and cognitive burden, ranging from cheap, noisy pairwise preference statements (PS) to richer but costlier indifference adjustments (IA). We study cost-aware optimization under an unknown scalarization and introduce QUIVER (Query-Informed Value Estimation for Regret), a surrogate-assisted evolutionary multi-objective optimizer that adaptively chooses between objective evaluations and heterogeneous preference queries. At each step, QUIVER selects the next action by maximizing the expected decision-quality improvement per unit total cost. Across DTLZ and WFG benchmarks under synthetic decision-maker models, QUIVER achieves the lowest final utility regret on challenging WFG problems (utility regret of 2.14 on WFG4, 2.82 on WFG9: a 25% improvement over baselines), outperforming all single-modality baselines. We analyze how the optimal mix of PS and IA adapts to problem difficulty: on easy problems (DTLZ2), QUIVER selects 80% PS queries; on hard problems (WFG9), it shifts to 35% IA queries. This adaptive modality selection demonstrates cost-aware preference learning in action.

Frontier AI systems perform best in settings with clear, stable, and verifiable objectives, such as code generation, mathematical reasoning, games, and unit-test-driven tasks. They remain less reliable in open-ended settings, including scientific assistance, long-horizon agents, high-stakes advice, personalization, and tool use, where the relevant objective is ambiguous, context-dependent, delayed, or only partially observable. We argue that many such failures are not merely failures of scale or capability, but failures of objective selection: the system optimizes a locally visible signal while missing which objectives should govern the interaction. We formulate this problem as \emph{contextual multi-objective optimization}. In this setting, systems must consider multiple, context-dependent objectives, such as helpfulness, truthfulness, safety, privacy, calibration, non-manipulation, user preference, reversibility, and stakeholder impact, while determining which objectives are active, which are soft preferences, and which must function as hard or quasi-hard constraints. These examples are not intended as an exhaustive taxonomy: different domains and deployment settings may activate different objective dimensions and different conflict-resolution procedures. Our framework models AI behavior as a context-dependent choice rule over candidate actions, objective estimates, active constraints, stakeholders, uncertainty, and conflict-resolution procedures. We outline an implementation pathway based on decomposed objective representations, context-to-objective routing, hierarchical constraints, deliberative policy reasoning, controlled personalization, tool-use control, diagnostic evaluation, auditing, and post-deployment revision.

We study day-ahead transmission topology control for high-voltage grid operation under $N-1$ security constraints. The operational task is to select, over a 24-hour horizon, a sequence of substation topologies obtained via busbar-coupler switching to relieve line overloads while limiting switching effort and topological complexity. We formulate this task as a sequential multi-objective optimization problem with four objectives used in TSO decision making: worst-case $N-1$ line loading, maximum topological depth, number of topology changes, and time spent outside the reference topology. We propose an exact block algorithm that exploits the temporal structure of topology plans: consecutive hours with the same topology are represented as blocks, enabling enumeration of the complete Pareto front over the admissible set of topologies under fixed operational bounds on depth and switching. We also develop a tailored NSGA-III-based evolutionary heuristic and evaluate it against the exact front. Using real operational data from the Dutch high-voltage transmission grid operated by TenneT, the block algorithm computes the exact front for a highly congested day in under three minutes after topology-level load-flow preprocessing. The exact front reveals low-switching plans with no DC $N-1$ thermal overloads that the tested evolutionary search fails to find. The proposed method, therefore, provides both a practical day-ahead decision-support tool for transmission operators and a benchmark for heuristic and learning-based topology-control methods.

This paper presents a novel population-based metaheuristic, Indian Wedding System Optimization (IWSO), inspired by the socio-cultural dynamics of traditional Indian weddings. IWSO models the matchmaking process driven by collaboration among families, candidates, and matchmakers as a guided, selective search framework for solving complex optimization problems. The algorithm introduces two key innovations: (i) a matchmaker-guided influence strategy, where elite solutions direct the evolution of weaker candidates, enhancing convergence without external parameters; and (ii) an adaptive elimination and reinitialization mechanism that maintains diversity and prevents premature convergence by replacing underperforming individuals. IWSO employs a weighted multi-objective fitness function and analytically derived time and space complexity, benchmarked against existing optimization approaches such as Genetic Algorithm (GA), Partical Swarm Optimization (PSO), Differential Evolution (DE), Cuckoo Search (CS), etc. Extensive experiments on benchmark high-dimensional and multimodal test functions demonstrate superior performance of IWSO in terms of convergence speed, solution quality, and robustness.

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.

Multiobjective optimization remains challenging for many scientific and engineering problems due to the need to balance convergence, diversity, and computational efficiency across high-dimensional objective landscapes. This work presents the Multiobjective Animorphic Ensemble Optimization (MAEO) framework, a parallelizable ensemble strategy that unifies state-of-the-art evolutionary algorithms within an island-based architecture, overcoming the limitations of relying on a single optimizer, as implied by the No Free Lunch theorem. MAEO uses a parameter-free hypervolume indicator for island performance assessment and a strict Pareto-rank-based individual scoring formulation that incorporates crowding distance and nadir-point proximity to ensure consistent selection pressure within each front. The framework is initiated using four algorithms (NSGA-III, CTAEA, AGEMOEA2, SPEA2) and evaluated through extensive benchmarking on 12 DTLZ/ZDT functions under 36 dimensionality settings using Wilcoxon signed-rank tests with both hypervolume and inverse generational distance metrics. Results show that MAEO achieves balanced convergence-diversity performance, outperforming or matching some of the leading multiobjective optimization algorithms across different benchmark problems. To demonstrate practical applicability, MAEO is applied to the equilibrium-cycle optimization of a small modular nuclear reactor. Eight discrete design variables (and three objectives (levelized cost of electricity, peak soluble boron concentration, fuel cycle length) are optimized under two safety constraints. The algorithm carried out roughly 40000 evaluations using computer simulations. MAEO identifies core designs that lower both the levelized cost of electricity and the peak boron concentration, while preserving fuel cycle length and meeting all safety constraints.

Large language models (LLMs) show promise for molecular optimization, but aligning them with selective and competing drug-design constraints remains challenging. We propose C-Moral, a reinforcement learning post-training framework for controllable multi-objective molecular optimization. C-Moral combines group-based relative optimization, property score alignment for heterogeneous objectives, and bottleneck-sensitive non-linear reward aggregation to improve stability across competing molecular properties. Experiments on C-MuMOInstruct and S$^2$-Bench MolOpt show that C-Moral achieves the best performance among compared methods on both benchmarks. On C-MuMOInstruct, C-Moral achieves the best Success Optimized Rate (SOR) of 48.9% on in-domain tasks and 39.5% on out-of-domain tasks while preserving scaffold similarity. On S$^2$-Bench MolOpt, it also achieves the strongest results across LogP, MR, and QED optimization tasks. These results suggest that C-Moral is an effective way to align molecular LLMs with continuous and constrained molecular design objectives. Our code and models are publicly available at https://github.com/Rwigie/C-MORAL.

Multi-task optimization is a powerful approach for solving a large number of tasks in parallel. However, existing algorithms face distinct limitations: Population-based methods scale poorly and remain underexplored for large task sets. Approaches that do scale beyond a thousand tasks are mostly MAP-Elites variants and rely on a fixed, discretized archive that disregards the topology of the task space. We introduce MONET (Multi-Task Optimization over Networks of Tasks), a multi-task optimization algorithm that models the task space as a graph: tasks are nodes, and edges connect tasks in the task parameter space. This representation enables knowledge transfer between tasks and remains tractable for high-dimensional problems while exploiting the topology of the task space. MONET combines social learning, which generates candidates from neighboring nodes via crossover, with individual learning, which refines a node's own solution independently via mutation. We evaluate MONET on four domains (archery, arm, and cartpole with 5,000 tasks each; hexapod with 2,000 tasks) and show that it matches or exceeds the performance of existing MAP-Elites-based baselines across all four domains.

Aligning large language models (LLMs) to desirable human values requires balancing multiple, potentially conflicting objectives such as helpfulness, truthfulness, and harmlessness, which presents a multi-objective optimisation challenge. Most alignment pipelines rely on a fixed scalarisation of these objectives, which can introduce procedural unfairness by systematically under-weighting harder-to-optimise or minority objectives. To promote more equitable trade-offs, we introduce MGDA-Decoupled, a geometry-based multi-objective optimisation algorithm that finds a shared descent direction while explicitly accounting for each objective's convergence dynamics. In contrast to prior methods that depend on reinforcement learning (e.g., GAPO) or explicit reward models (e.g., MODPO), our approach operates entirely within the lightweight Direct Preference Optimisation (DPO) paradigm. Experiments on the UltraFeedback dataset show that geometry-aware methods -- and MGDA-Decoupled in particular -- achieve the highest win rates against golden responses, both overall and per objective.

Flight diversions are rare but high-impact events in aviation, making their reliable prediction vital for both safety and operational efficiency. However, their scarcity in historical records impedes the training of machine learning models utilised to predict them. This study addresses this scarcity gap by investigating how generative models can augment historical flight data with synthetic diversion records to enhance model training and improve predictive accuracy. We propose a multi-objective optimisation framework coupled with automated hyperparameter search to identify optimal configurations for three deep generative models: Tabular Variational Autoencoder (TVAE), Conditional Tabular Generative Adversarial Network (CTGAN), and CopulaGAN, with the Gaussian Copula (GC) model serving as a statistical baseline. The quality of the synthetic data was examined through a six-stage evaluation framework encompassing realism, diversity, operational validity, statistical similarity, fidelity, and predictive utility. Results show that the optimised models significantly outperform their non-optimised counterparts, and that synthetic augmentation substantially improves diversion prediction compared to models trained solely on real data. These findings demonstrate the effectiveness of hyperparameter-optimised generative models for advancing predictive modelling of rare events in air transportation.

Generative answer engines expose content through selective citation rather than ranked retrieval, fundamentally altering how visibility is determined. This shift calls for new optimization methods beyond traditional search engine optimization. Existing generative engine optimization (GEO) approaches primarily rely on token-level text rewriting, offering limited interpretability and weak control over the trade-off between citation visibility and content quality. We propose FeatGEO, a feature-level, multi-objective optimization framework that abstracts webpages into interpretable structural, content, and linguistic properties. Instead of directly editing text, FeatGEO optimizes over this feature space and uses a language model to realize feature configurations into natural language, decoupling high-level optimization from surface-level generation. Experiments on GEO-Bench across three generative engines demonstrate that FeatGEO consistently improves citation visibility while maintaining or improving content quality, substantially outperforming token-level baselines. Further analyses show that citation behavior is more strongly influenced by document-level content properties than by isolated lexical edits, and that the learned feature configurations generalize across language models of different scales.

This paper proposes a multi-step probabilistic forecasting framework using a single neural-network based model to generate simultaneous point and interval forecasts. Our approach ensures non-crossing prediction intervals (PIs) through a model structure design that strictly satisfy a target coverage probability (PICP) while maximizing sharpness. Unlike existing methods that rely on manual weight tuning for scalarized loss functions, we treat point and PI forecasting as a multi-objective optimization problem, utilizing multi-gradient descent to adaptively select optimal weights. Key innovations include a new PI loss function based on an extended log-barrier with an adaptive hyperparameter to guarantee the coverage, a hybrid architecture featuring a shared temporal model with horizon-specific submodels, and a training strategy. The proposed loss is scale-independent and universally applicable; combined with our training algorithm, the framework eliminates trial-and-error hyperparameter tuning for balancing multiple objectives. Validated by an intra-day solar irradiance forecasting application, results demonstrate that our proposed loss consistently outperforms those in current literature by achieving target coverage with the narrowest PI widths. Furthermore, when compared against LSTM encoder-decoder and Transformer architectures--including those augmented with Chronos foundation models--our method remains highly competitive and can be seamlessly adapted to any deep learning structure.

Due to the increasing frequency and severity of storm events, driven by the escalation of anthropogenic climate change and urban expansion, there is a requirement for increasingly efficient flood risk management strategies. While Blue-Green Infrastructure (BGI) offers a sustainable solution for managing flood risk, optimal implementation is challenging. To help overcome this challenge, this study presents a novel multi-objective optimisation tool that couples a state-of-the-art hydrodynamic model with a bespoke evolutionary algorithm. The use of a fully dynamic hydrodynamic model enables the tool to accurately evaluate the effectiveness of proposed BGI features with respect to property scale flood vulnerability and hazard analysis. This contrasts with alternative approaches which utilise simplified models, which can only reliably predict inundation extents, thus the proposed optimisation tool provides greater certainty regarding the optimality of the solutions. As a hydrodynamic simulation is required to evaluate each candidate solution, the bespoke evolutionary algorithm is specifically designed to minimise the number of simulations required, ensuring the tool is computationally practical. The effectiveness of the tool in this regard is validated via the derivation of exact convergence measures, for a tractable search space, and via comparisons with benchmark algorithms, for an intractable search space. Compared with traditional design practices, the proposed tool offers an automated approach capable of efficiently exploring a wide range of solutions, providing decision-makers with a set of optimal solutions from which they can make informed investment decisions. The presented methods provide a robust framework for optimising a variety of BGI features in complex urban environments.

Multi-objective Bayesian optimization (MOBO) provides a principled framework for optimizing multiple expensive black-box functions. However, existing MOBO methods often struggle with coverage, scalability, and handling constraints and preferences. In this work we propose STAGE-BO, Sequential Targeting Adaptive Gap-Filling $\varepsilon$-Constraint Bayesian Optimization: by analyzing the coverage of the surrogate Pareto front, our method identifies the Pareto front point with the largest uncovered gap, and uses its coordinates to define adaptive constraints in $\varepsilon$-constraint method, which transforms the problem into a sequence of inequality-constrained subproblems, efficiently solved via constrained expected improvement acquisition. Our approach provides uniform Pareto coverage without hypervolume computation and naturally handles constraints and preferences. Experiments on synthetic and real-world benchmarks demonstrate superior coverage and competitive hypervolume performance against state-of-the-art baselines. Our code implementation can be found at https://github.com/YangYaohong1/STAGE-BO.

Pareto optimization via evolutionary multi-objective algorithms has been shown to efficiently solve constrained monotone submodular functions. Traditionally when solving multiple problems, the algorithm is run for each problem separately. We introduce multitasking formulations of these problems that are an effective way to solve multiple related problems with a single run. In our setting the given problems share a monotone submodular function $f$ but have different knapsack constraints. We examine the case where elements within a constraint have the same cost and show that our multitasking formulations result in small Pareto fronts. This allows the population to share solutions between all problems leading to significant improvements compared to running several classical approaches independently. Using rigorous runtime analysis, we analyze the expected time until the introduced multitasking approaches obtain a $(1-1/e)$-approximation for each of the given problems. Our experimental investigations for the maximum coverage problem give further insight into the dynamics behind how the approach works and doesn't work in practice for problems where elements within a constraint also have varied costs.

Open RAN (O-RAN) exposes rich control and telemetry interfaces across the Non-RT RIC, Near-RT RIC, and distributed units, but also makes it harder to operate multi-tenant, multi-objective RANs in a safe and auditable manner. In parallel, agentic AI systems with explicit planning, tool use, memory, and self-management offer a natural way to structure long-lived control loops. This article surveys how such agentic controllers can be brought into O-RAN: we review the O-RAN architecture, contrast agentic controllers with conventional ML/RL xApps, and organise the task landscape around three clusters: network slice life-cycle, radio resource management (RRM) closed loops, and cross-cutting security, privacy, and compliance. We then introduce a small set of agentic primitives (Plan-Act-Observe-Reflect, skills as tool use, memory and evidence, and self-management gates) and show, in a multi-cell O-RAN simulation, how they improve slice life-cycle and RRM performance compared to conventional baselines and ablations that remove individual primitives. Security, privacy, and compliance are discussed as architectural constraints and open challenges for standards-aligned deployments. This framework achieves an average 8.83% reduction in resource usage across three classic network slices.

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.

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.

Multi-task optimization is typically characterized by a fixed and finite set of tasks. The present paper relaxes this condition by considering a non-fixed and potentially infinite set of optimization tasks defined in a parameterized, continuous and bounded task space. We refer to this unique problem setting as parametric multi-task optimization (PMTO). Assuming the bounds of the task parameters to be ($\boldsymbolθ_l$, $\boldsymbolθ_u$), a novel ($\boldsymbolθ_l$, $\boldsymbolθ_u$)-PMTO algorithm is crafted to operate in two complementary modes. In an offline optimization mode, a joint search over solution and task spaces is carried out with the creation of two approximation models: (1) for mapping points in a unified solution space to the objective spaces of all tasks, which provably accelerates convergence by acting as a conduit for inter-task knowledge transfers, and (2) for probabilistically mapping tasks to their corresponding solutions, which facilitates evolutionary exploration of under-explored regions of the task space. In the online mode, the derived models enable direct optimization of any task within the bounds without the need to search from scratch. This outcome is validated on both synthetic test problems and practical case studies, with the significant real-world applicability of PMTO shown towards fast reconfiguration of robot controllers under changing task conditions. The potential of PMTO to vastly speedup the search for solutions to minimax optimization problems is also demonstrated through an example in robust engineering design.

In recent years, explainability in machine learning has gained importance. In this context, counterfactual explanation (CE), which is an explanation method that uses examples, has attracted attention. However, it has been pointed out that CE is not robust when there are multiple machine-learning models with similar accuracy. These problems are important when using machine learning to make safe decisions. In this paper, we propose robust CEs that introduce a new viewpoint -- Pareto improvement -- and a method that uses multi-objective optimization to generate it. To evaluate the proposed method, we conducted experiments using both simulated and real data. The results demonstrate that the proposed method is both robust and practical. This study highlights the potential of ensuring robustness in decision-making by applying the concept of social welfare. We believe that this research can serve as a valuable foundation for various fields, including explainability in machine learning, decision-making, and action planning based on machine learning.