Multi-Objective Evolutionary Optimization

This work presents a novel variant of the Firefly Algorithm (FA) for data clustering, addressing limitations of traditional methods like K-Means that struggle with non-uniform cluster shapes, densities, and the need for pre-defining the number of clusters. The proposed algorithm introduces a centroid movement strategy and a multi-objective fitness function that balances compactness, separation, and a novel TSP-based navigation penalty. It automatically estimates the optimal number of clusters and dynamically adjusts cluster boundaries. Application to robotic sensor networks highlights its practical value, with experiments showing improved clustering quality and reduced intra-cluster path distances compared to K-Means. These results confirm the algorithm's robustness in complex spatial clustering tasks, with potential for future extensions to higher-dimensional and adaptive scenarios.

Multimodal optimization requires finding many optima rather than merely keeping a diverse population. Yet most niching-based evolutionary algorithms rely on distances or density estimators without explicitly recovering the underlying peak--basin organization in the decision space, which can lead to pseudo-multimodality: many distinct individuals ultimately collapse into only a few basins. We introduce Chaotic Landscape-Decoding Evolution (CLDE), a decision-space-centric framework that turns multimodal search into a closed loop of decode--value--allocate--refine. CLDE injects controlled global exploration via a logistic chaotic map with a decaying step size, then builds a $k$-nearest-neighbor graph on a decoding canvas and performs persistence-guided basin growing that merges peaks only when they are not separated by deep valleys. An adaptive persistence threshold continuously tunes the decoding resolution online to avoid over-fragmentation and over-merging. Guided by the decoded structure, CLDE carries out basin-wise selection and refinement to improve solution quality while preserving basin coverage. We instantiate CLDE as CLDE-S and CLDE-M for single- and multi-objective multimodal optimization. Experiments on 20 CEC2013 functions show that CLDE-S achieves strong peak ratio under the same evaluation budget, while on DTLZ and MMMOP suites CLDE-M attains competitive IGD/IGDx, with pronounced gains on strongly multimodal problems.

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.

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.

Advances in large language models (LLMs) have recently opened new and promising avenues for small-molecule drug discovery. Yet existing LLM-based approaches for molecular generation often suffer from high rates of invalid and low-quality ligand candidates, a result of the syntactic limitations of current models with regard to molecular strings. In this paper, we introduce $\texttt{ToolMol}$, an evolutionary agentic framework for de novo drug design. $\texttt{ToolMol}$ combines a multi-objective genetic algorithm with an agentic LLM operator that iteratively updates the ligand population. We build a comprehensive toolbox of RDKit-backed functions that allows our agentic operator to consisently make precise ligand modifications. $\texttt{ToolMol}$ achieves state-of-the-art performance on multi-objective property optimization tasks, discovering drug-like and synthesizable ligands that have $>10\%$ stronger predicted binding affinity compared to existing methods, evaluated on three protein targets. $\texttt{ToolMol}$ ligands additionally achieve state-of-the-art results in gold-standard Absolute Binding Free Energy scores, gaining over existing methods by over $35\%$. By studying chain-of-thought reasoning traces, we observe that tool-calling enables the model to more faithfully execute its planned modifications, efficiently exploiting the strong chemical prior knowledge in LLMs.

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.

Gradient-based preference optimization methods for large language model (LLM) alignment suffer from preference collapse, converging to narrow behavioral modes while neglecting preference diversity. We introduce EvoPref, a multi-objective evolutionary algorithm that maintains populations of Low-Rank Adaptation (LoRA) adapters optimized across helpfulness, harmlessness, and honesty objectives using Non-dominated Sorting Genetic Algorithm II (NSGA-II) selection with archive-based diversity preservation. Our primary contribution is demonstrating that population-based methods discover substantially more diverse alignments than gradient descent. On standard benchmarks, EvoPref improves preference coverage by 18% (median 82.5% vs. 70.0% for ORPO, $p<0.001$, Wilcoxon, $n=30$) and reduces collapse rates by 47% (11.0% vs. 20.6%, $p<0.001$), while achieving competitive alignment quality (median 75.5% RewardBench vs. 75.0% for ORPO, $p<0.05$). We provide theoretical motivation extending recent multi-objective evolutionary algorithm (MOEA) runtime analysis (Dang et al., 2025) suggesting why archive-based methods escape collapse more effectively than single-trajectory optimization. Comprehensive comparisons against MOEA/D, SMS-EMOA, CMA-ES, and gradient baselines (DPO, IPO, KTO, ORPO) with rigorous statistical testing (Friedman with Holm correction, Vargha-Delaney effect sizes, median with IQR) confirm that multi-objective selection with diversity preservation is essential. This work establishes evolutionary optimization as a principled paradigm for diverse LLM alignment.

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.

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.

Stopping criteria automatically determine when to stop an evolutionary algorithm, so as not to waste function evaluations on a stagnant population. Although stopping criteria play an important role in real-world applications, they have attracted little attention in the evolutionary multi-objective optimization (EMO) community. In fact, new stopping criteria for EMO have been rarely developed in recent years. One reason for the stagnation in developing stopping criteria for EMO is a lack of effective benchmarking methodologies. To address this issue, this paper proposes (i) a performance measure of stopping criteria for EMO and (ii) a file-based benchmarking approach. This paper also proposes (iii) a data representation method that effectively stores population states in text files. (i) The proposed measure represents the performance of stopping criteria as a single scalar value, making comparison easy. (ii) The proposed file-based approach not only simplifies the benchmarking process but also facilitates reproducibility. (iii) The proposed data representation method addresses the issue of file size in (ii). We demonstrate the effectiveness of our three contributions (i)--(iii) by benchmarking five representative stopping criteria for EMO.

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.

Scalability of evolutionary algorithms refers to assessing how their performance changes as problem size increases. In the area of multi-objective optimisation, research on the scalability of multi-objective evolutionary algorithms (MOEAs) has predominantly focussed on continuous problems. However, multi-objective combinatorial optimisation problems (MOCOPs) differ from continuous ones. Their discrete and rigid structure often brings rugged landscape, numerous local optimal solutions and disjoint global optimal regions. This leads to different behaviour of MOEAs. For example, SEMO, a simple MOEA without mating selection and diversity maintenance mechanisms, has been shown to be highly competitive, and in many cases to outperform more sophisticated MOEAs on MOCOPs. Yet, it remains unclear whether such findings hold for large-scale cases. In this paper, we conduct an empirical investigation into the scalability of MOEAs on combinatorial problems, with problem size from 50 to 5,000. Our results show that SEMO experiences a decline in convergence speed as dimensionality increases, compared to other MOEAs such as NSGA-II, SMS-EMOA and MOEA/D. We further demonstrate that the absence of crossover is a major contributor to SEMO's underperformance in large-scale problems, and that incorporating crossover into SEMO can substantially accelerate convergence in general, despite being detrimental in spreading solutions over the Pareto front.

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.

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.

LLM-based RTL code generation methods increasingly target both functional correctness and PPA quality, yet existing approaches universally decouple the two objectives, optimizing PPA only after correctness is fully achieved. Whether through sequential multi-agent pipelines, evolutionary search with binary correctness gates, or hierarchical reward dependencies, partially correct but architecturally promising candidates are systematically discarded. Moreover, existing methods reduce the multi-objective PPA space to a single scalar fitness, obscuring the trade-offs among area, delay, and power. To address these limitations, we propose COEVO, a co-evolutionary framework that unifies correctness and PPA optimization within a single evolutionary loop. COEVO formulates correctness as a continuous co-optimization dimension alongside area, delay, and power, enabled by an enhanced testbench that provides fine-grained scoring and detailed diagnostic feedback. An adaptive correctness gate with annealing allows PPA-promising but partially correct candidates to guide the search toward jointly optimal solutions. To preserve the full PPA trade-off structure, COEVO employs four-dimensional Pareto-based non-dominated sorting with configurable intra-level sorting, replacing scalar fitness without manual weight tuning. Evaluated on VerilogEval 2.0 and RTLLM 2.0, COEVO achieves 97.5% and 94.5% Pass@1 with GPT-5.4-mini, surpassing all agentic baselines across four LLM backbones, while attaining the best PPA on 43 out of 49 synthesizable RTLLM designs.

Open-pit mine scheduling is a complex real-world optimization problem that involves uncertain economic values and dynamically changing resource capacities. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments. However, uncertainty and dynamic changes are often studied in isolation in real-world problems. In this paper, we study a dynamic chance-constrained open-pit mine scheduling problem in which block economic values are stochastic and mining and processing capacities vary over time. We adopt a bi-objective evolutionary formulation that simultaneously maximizes expected discounted profit and minimizes its standard deviation. To address dynamic changes, we propose a diversity-based change response mechanism that repairs a subset of infeasible solutions and introduces additional feasible solutions whenever a change is detected. We evaluate the effectiveness of this mechanism across four multi-objective evolutionary algorithms and compare it with a baseline re-evaluation-based change-response strategy. Experimental results on six mining instances demonstrate that the proposed approach consistently outperforms the baseline methods across different uncertainty levels and change frequencies.

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.