Multi-objective optimization problems (MOPs) require the simultaneous optimization of conflicting objectives. Real-world MOPs often exhibit complex characteristics, including high-dimensional decision spaces, many objectives, or computationally expensive evaluations. While population-based evolutionary computation has shown promise in addressing diverse MOPs through problem-specific adaptations, existing approaches frequently lack generalizability across distinct problem classes. Inspired by pre-training paradigms in machine learning, we propose a Population Pre-trained Model (PPM) that leverages historical optimization knowledge to solve complex MOPs within a unified framework efficiently. PPM models evolutionary patterns via population modeling, addressing two key challenges: (1) handling diverse decision spaces across problems and (2) capturing the interdependency between objective and decision spaces during evolution. To this end, we develop a population transformer architecture that embeds decision spaces of varying scales into a common latent space, enabling knowledge transfer across diverse problems. Furthermore, our architecture integrates objective-space features through objective fusion to enhance population prediction accuracy for complex MOPs. Our approach achieves robust generalization to downstream optimization tasks with up to 5,000 dimensions-five times the training scale and 200 times greater than prior work. Extensive evaluations on standardized benchmarks and out-of-training real-world applications demonstrate the consistent superiority of our method over state-of-the-art algorithms tailored to specific problem classes, improving the performance and generalization of evolutionary computation in solving MOPs.
Sequential perceptual aliasing is a cognitive challenge for learning agents when robots cannot differentiate states and their associations based on immediate observations, leading to poor decision-making. Existing systems struggle to abstract and distinguish observations effectively to achieve policy learning. This paper addresses this issue by introducing new aliasing types within the context of sequential aliasing and proposing an enhanced XCS classifier system that learns using a complete state-action map. The proposed system called hierarchical Frames-of-References-based XCS (Hi-FoRsXCS), can concatenate sequences of aliased states with the same observation into a chain. Hi- FoRsXCS then predicts associations between the observations and aliased states using the ends of the chain, enabling optimal policy learning with a complete action map. Experimental results demonstrate that Hi-FoRsXCS outperforms the existing systems in terms of accuracy. However, the limitations of Hi-FoRsXCS will be discussed in this paper.
The main challenge in handling dynamic multi-objective optimization problems lies in the need for algorithms to accurately track Pareto-optimal solutions in constantly changing environments. Most existing predictionbased dynamic multi-objective evolutionary algorithms (DMOEAs) conduct prediction either in the decision space or the objective space alone, or apply the same prediction model to both spaces. However, such approaches may fail to fully capture the distinct change patterns of each space, especially under nonlinear and complex environmental dynamics, thereby limiting the effectiveness of these algorithms. Furthermore, when sampling methods are used to help the algorithm generate populations in new environments, a large number of sampled individuals can impose a significant computational burden due to the increased number of function evaluations. To address these limitations, this paper proposes a dynamic multi-objective evolutionary algorithm, namely DS-DMOEA, which efficiently adapts to environmental changes through a dual-space prediction strategy and a surrogate-based sampling strategy. The dual-space prediction strategy captures dynamic changes by employing a weight vector-based method in the objective space and a geodesic flow kernel method in the decision space. Simultaneously, the surrogate-based sampling strategy generates a high-quality sampling population by training surrogate models with information from similar historical environments. The predicted and sampled populations are then combined to form an initial population well-suited for the new environment. DS-DMOEA has been tested against nine state-of-the-art DMOEAs on 19 benchmark problems with three types of environmental change patterns. The experimental results validate the effectiveness of the proposed algorithm.
Ill-conditioned problems are widely acknowledged as a major challenge in singleobjective optimization, yet they remain largely unexplored in evolutionary multiobjective optimization. In this paper, we introduce a decomposition-based multiobjective evolution strategy (MOES/D) for optimizing non-separable and ill-conditioned multiobjective problems. In contrast to most existing approaches that integrate evolution strategies while potentially compromising their essential features, we develop novel, tailored strategies to coordinate evolution strategies, maximizing their strengths. These strategies collectively contribute to the efficiency of MOES/D, which include an importance mixing algorithm that enhances sample efficiency in an unbiased manner, a collaborative ascent method that optimizes multiple subproblems simultaneously, and a principled resource allocation based on expectation-maximization that prioritizes the evolution strategy models. To bridge the gap in the field, we propose a novel benchmark suite in which all instances are non-separable and either moderate- or illconditioned. Extensive experiments on the suite demonstrate that MOES/D excels at solving moderate- or ill-conditioned multiobjective problems, outperforming most state-of-the-art algorithms by a significant margin.
We introduce and explore the automation and adaptation of partial neighborhood local search. Unlike traditional approaches requiring extensive parameter tuning, we design our approach to operate with minimal prerequisites. Specifically, we extend the sampled walk and ID walk algorithms by using distance-based calculations over a sliding window to determine the number of neighbors to evaluate at each step. To validate their performance, we empirically evaluate these parameter-free methods on four challenging combinatorial optimization benchmark problem classes from the literature, comparing them against fixed-parameter versions across multiple values. Our experiments show that, despite their simplicity, generic nature, and absence of parameters, these approaches achieve robust and competitive results across diverse problems-including different solution representations, neighborhood structures, and fitness landscape characteristics-thus validating the viability of generic autonomous local search methods.
Experimental robot optimization often requires evaluating each candidate policy for seconds to minutes. The chosen evaluation time influences optimization because of a speed-accuracy tradeoff: shorter evaluations enable faster iteration, but are also more subject to noise. Here, we introduce a supplement to the CMA-ES optimization algorithm, named Adaptive Sampling CMA-ES (AS-CMA), which assigns sampling time to candidates based on predicted sorting difficulty, aiming to achieve consistent precision. We compared AS-CMA to CMA-ES and Bayesian optimization using a range of static sampling times in four simulated cost landscapes. AS-CMA converged on 98% of all runs without adjustment to its tunable parameter, and converged 24-65% faster and with 29-76% lower total cost than each landscape's best CMA-ES static sampling time. As compared to Bayesian optimization, AS-CMA converged more efficiently and reliably in complex landscapes, while in simpler landscapes, AS-CMA was less efficient but equally reliable. We deployed AS-CMA in an exoskeleton optimization experiment and found the optimizer's behavior was consistent with expectations. These results indicate that AS-CMA can improve optimization efficiency in the presence of noise while minimally affecting optimization setup complexity and tuning requirements.
To address optimization problems that involve expensive evaluations with unknown and heterogeneous costs, cost-aware Bayesian optimization (BO) emerges as a prominent solution in many real-world scenarios. However, as a critical step in developing BO algorithms, the design of efficient cost-aware acquisition functions (AFs) remains a significant challenge. This paper introduces EvolCAF, a novel framework that integrates large language models (LLMs) with evolutionary computation (EC) to automatically design cost-aware AFs. Leveraging the crossover and mutation in the algorithmic space, EvolCAF offers a novel design fashion, significantly reducing the reliance on domain expertise and labor-intensive trial-and-error process in the traditional manual design paradigm. We find the best AF designed by EvolCAF effectively utilizes the available information, including historical data, surrogate models and budget details. It introduces novel ideas not previously explored in the existing literature on acquisition function design, allowing for clear interpretations to provide insights into its behavior and decision-making process. In comparison to the well-known EIpu and EI-cool methods designed by human experts, our approach showcases remarkable efficiency and generalization across various synthetic and real-world tasks.
Designing algorithms for optimization problems, no matter heuristic or metaheuristic, often relies on manual design and domain expertise, limiting their scalability and adaptability. The integration of Large Language Models (LLMs) and Evolutionary Algorithms (EAs) presents a promising new way to overcome these limitations to make optimization be more automated, where LLMs function as dynamic agents capable of generating, refining, and interpreting optimization strategies, while EAs explore complex searching spaces efficiently through evolutionary operators. Since this synergy enables a more efficient and creative searching process, we first review important developments in this direction, and then summarize an LLM-EA paradigm for automated optimization algorithm design. We conduct an in-depth analysis on innovative methods for four key EA modules, namely, individual representation, selection, variation operators, and fitness evaluation, addressing challenges related to optimization algorithm design, particularly from the perspective of LLM prompts, analyzing how the prompt flows evolving with the evolutionary process, adjusting based on evolutionary feedback (e.g., population diversity, convergence rate). Furthermore, we analyze how LLMs, through flexible prompt-driven roles, introduce semantic intelligence into fundamental EA characteristics, including diversity, convergence, adaptability, and scalability. Our systematic review and thorough analysis into the paradigm can help researchers better understand the current research and boost the development of synergizing LLMs with EAs for automated optimization algorithm design.
Mixed-integer (MI) quadratic models subject to quadratic constraints, known as All-Quadratic MI Programs, constitute a challenging class of NP-complete optimization problems. The particular scenario of unbounded integers defines a subclass that holds the distinction of being even undecidable. This complexity suggests a possible soft-spot for mathematical programming (MP) techniques, which otherwise constitute a good choice to treat MI problems. We consider the task of minimizing MI convex quadratic objective and constraint functions with unbounded decision variables. Given the theoretical weakness of white-box MP solvers to handle such models, we turn to black-box meta-heuristics of the evolution strategies (ESs) family, and question their capacity to solve this challenge. Through an empirical assessment of all-quadratic test-cases, across varying Hessian forms and condition numbers, we compare the performance of the CPLEX solver to modern MI ESs, which handle constraints by penalty. Our systematic investigation begins where the CPLEX solver encounters difficulties (timeouts as the search-space dimensionality increases, D≳30), and we report in detail on the D=64 case. Overall, the empirical observations confirm that black-box and white-box solvers can be competitive over this MI problem class, exhibiting 67% similar performance in terms of the attained objective function values in a fixed-budget perspective. Despite consistent termination in timeouts, CPLEX demonstrated superior or comparable performance to the MIESs in 98% of the cases. This trend is flipped when unboundedness is amplified by a significant translation of the optima, leading to a totally inferior performance of CPLEX across 81% of the cases. We also conclude that conditioning and separability are not intuitive factors in determining the hardness degree of this MI problem class.
Extending state-of-the-art evolutionary algorithms is a widespread research direction. This trend has resulted in algorithms that give good results but are complex and challenging to analyze. One of these algorithms is EA4Eig-the winner of the CEC 2022 competition on single objective bound constrained search. The algorithm internally uses four optimization algorithms with modified components. This paper presents an analysis of EA4Eig and proposes a simplified version thereof exhibiting better optimization performance. The analysis found that the original source code contains errors that impact the algorithm's rank. The code was corrected, and the CEC 2022 competition ranking was recalculated. The impact of individual EA4Eig components on its performance was empirically analyzed. As a result, the algorithm was simplified by removing two of the components. The best remaining component was analyzed further, which made it possible to remove some unnecessary and harmful code. Several versions of the algorithm were created and tested, varying in the degree of simplification. The simplest of them is implemented in 244 lines of C++ code, whereas the original implementation used 716 lines of Matlab code. Further analyses focused on the parameters of the algorithm. The constants hidden in the source code were named and treated as additional configurable parameters that underwent tuning. The ablation analyses showed that two of these hidden parameters had the most significant impact on the improvement achieved by the tuned version. The results of the original and simplified versions were compared on CEC 2022 and BBOB benchmarks. The results confirm that the simplified version is better than the original one on both these benchmarks.
Dynamic flexible job shop scheduling (DFJSS) is an important combinatorial optimisation problem, requiring simultaneous decision-making for machine assignment and operation sequencing in dynamic environments. Genetic programming (GP), as a hyper-heuristic approach, has been extensively employed for acquiring scheduling heuristics for DFJSS. A drawback of GP for DFJSS is that GP has weak exploration ability indicated by its quick diversity loss during the evolutionary process. This paper proposes an effective GP algorithm with tabu lists to capture the information of explored areas and guide GP to explore more unexplored areas to improve GP's exploration ability for enhancing GP's effectiveness. First, we use phenotypic characterisation to represent the behaviour of tree-based GP individuals for DFJSS as vectors. Then, we build tabu lists that contain phenotypic characterisations of explored individuals at the current generation and across generations, respectively. Finally, newly generated offspring are compared with the individuals' phenotypic characterisations in the built tabu lists. If an individual is unseen in the tabu lists, it will be kept to form the new population at the next generation. Otherwise, it will be discarded. We have examined the proposed GP algorithm in nine different scenarios. The findings indicate that the proposed algorithm outperforms the compared algorithms in the majority of scenarios. The proposed algorithm can maintain a diverse and well-distributed population during the evolutionary process of GP. Further analyses show that the proposed algorithm does cover a large search area to find effective scheduling heuristics by focusing on unseen individuals.
One of the most common problems in data mining applications is the uneven distribution of classes, which appears in many real-world scenarios. The class of interest is often highly underrepresented in the given dataset, which harms the performance of most classifiers. One of the most successful methods for addressing the class imbalance problem is to oversample the minority class using synthetic samples. Since the original algorithm, the synthetic minority oversampling technique (SMOTE), introduced this method, numerous versions have emerged, each of which is based on a specific hypothesis about where and how to generate new synthetic instances. In this paper, we propose a different approach based exclusively on evolutionary computation that imposes no constraints on the creation of new synthetic instances. Majority class undersampling is also incorporated into the evolutionary process. A thorough comparison involving three classification methods, 85 datasets, and more than 90 class-imbalance strategies shows the advantages of our proposal.
Most many-objective optimization algorithms (MaOEAs) adopt a pre-assumed Pareto front (PF) shape, instead of the true PF shape, to balance convergence and diversity in high-dimensional objective space, resulting in insufficient selection pressure and poor performance. To address these shortcomings, we propose MaOEA-PV based on PF shape classification and vector angle selection. The three innovation points of this paper are as follows: (i) a new method for PF classification; (ii) a new fitness function that combines convergence and diversity indicators, thereby enhancing the quality of parents during mating selection; and (iii) the selection of individuals exhibiting the best convergence to add to the population, overcoming the lack of selection pressure during environmental selection. Subsequently, the max-min vector angle strategy is employed. The solutions with the highest diversity and the least convergence are selected based on the max and min vector angles, respectively, which balances convergence and diversity. The performance of algorithm is compared with those of five state-of-the-art MaOEAs on 41 test problems and 5 real-world problems comprising as many 15 objectives. The experimental results demonstrate the competitive and effective nature of the proposed algorithm.
Automated algorithm configuration aims at finding well-performing parameter configurations for a given problem, and it has proven to be effective within many AI domains, including evolutionary computation. Initially, the focus was on excelling in one performance objective, but, in reality, most tasks have a variety of (conflicting) objectives. The surging demand for trustworthy and resource-efficient AI systems makes this multiobjective perspective even more prevalent. We propose a new general-purpose multiobjective automated algorithm configurator by extending the widely-used SMAC framework. Instead of finding a single configuration, we search for a nondominated set that approximates the actual Pareto set. We propose a pure multiobjective Bayesian optimization approach for obtaining promising configurations by using the predicted hypervolume improvement as acquisition function. We also present a novel intensification procedure to efficiently handle the selection of configurations in a multiobjective context. Our approach is empirically validated and compared across various configuration scenarios in four AI domains, demonstrating superiority over baseline methods, competitiveness with MO-ParamILS on individual scenarios, and an overall best performance.
The representation of optimization problems and algorithms in terms of numerical features is a well-established tool for comparing optimization problem instances, for analyzing the behavior of optimization algorithms, and the quality of existing problem benchmarks, as well as for automated per-instance algorithm selection and configuration approaches. Extending purely problem-centered feature collections, our recently proposed DynamoRep features provide a simple and inexpensive representation of the algorithmproblem interaction during the optimization process. In this paper, we conduct a comprehensive analysis of the predictive power of the DynamoRep features for the problem classification, algorithm selection, and algorithm classification tasks. In particular, the features are evaluated for the classification of problem instances into problem classes from the BBOB (Black Box Optimization Benchmarking) suite, selecting the best algorithm to solve a given problem from a portfolio of three algorithms (Differential Evolution, Evolutionary Strategy, and Particle Swarm Optimization), as well as distinguishing these algorithms based on their trajectories. We show that, despite being much cheaper to compute, they can yield results comparable to those using state-ofthe-art Exploratory Landscape Analysis features.
Interactive methods support decision-makers in finding the most preferred solution for multiobjective optimization problems, where multiple conflicting objective functions must be optimized simultaneously. These methods let a decision-maker provide preference information iteratively during the solution process to find solutions of interest, allowing them to learn about the trade-offs in the problem and the feasibility of the preferences. Several interactive evolutionary multiobjective optimization methods have been proposed in the literature. In the evolutionary computation community, the so-called decomposition-based methods have been increasingly popular because of their good performance in problems with many objective functions. They decompose the multiobjective optimization problem into multiple subproblems to be solved collaboratively. Various interactive versions of decomposition-based methods have been proposed. However, most of them do not consider the desirable properties of real interactive solution processes, such as avoiding imposing a high cognitive burden on the decision-maker, allowing them to decide when to interact with the method, and supporting them in selecting a final solution. This article reviews interactive evolutionary decomposition-based multiobjective optimization methods and different methodologies utilized to incorporate interactivity in them. Additionally, desirable properties of interactive decomposition-based multiobjective evolutionary optimization methods are identified, aiming to make them easier to be applied in real-world problems.
Decomposition-based multiobjective evolutionary algorithms (MOEAs) are popular methods utilized to address many-objective optimization problems (MaOPs). These algorithms decompose the original MaOP into several scalar optimization subproblems, and solve them to obtain a set of solutions to approximate the Pareto front (PF). The decomposition approach is an important component in them. This paper presents a runtime analysis of a MOEA based on the classic decomposition framework using the typical weighted sum (WS), Tchebycheff (TCH), and penalty-based boundary intersection (PBI) approaches to obtain an optimal solution for any subproblem of two pseudo-Boolean benchmark MaOPs, namely mLOTZ and mCOCZ. Due to the complexity and limitation of the theoretical analysis techniques, the analyzed algorithm employs 1-bit mutation to generate offspring individuals. The results indicate that when using WS, the analyzed algorithm can consistently find an optimal solution for every subproblem, which is located in the PF, in polynomial expected runtime. In contrast, the algorithm requires at least exponential expected runtime (with respect to the number of objectives m) for certain subproblems when using TCH or PBI, even though the landscapes of all objective functions in the two benchmarks are strictly monotone. Moreover, this analysis reveals a drawback of using WS: the optimal solutions obtained by solving subproblems are more easily mapped to the same point in the PF, compared to the case of using TCH. When using PBI, a smaller value of the penalty parameter is a good choice for faster convergence to the PF but may compromise diversity. To further understand the impact of these approaches in practical algorithms, numerical experiments on using bit-wise mutation to generate offspring individuals are conducted. The findings of this study may be helpful for designing more efficient decomposition approaches for MOEAs in future research.
High-dimensionality is one of the serious real-world data challenges in symbolic regression and it is more challenging if the data are incomplete. Genetic programming has been successfully utilised for high-dimensional tasks due to its natural feature selection ability, but it is not directly applicable to incomplete data. Commonly, it needs to impute the missing values first and then perform genetic programming on the imputed complete data. However, in the case of having many irrelevant features being incomplete, intuitively, it is not necessary to perform costly imputations on such features. For this purpose, this work proposes a genetic programming-based approach to select features directly from incomplete high-dimensional data to improve symbolic regression performance. We extend the concept of identity/neutral elements from mathematics into the function operators of genetic programming, thus they can handle the missing values in incomplete data. Experiments have been conducted on a number of data sets considering different missingness ratios in high-dimensional symbolic regression tasks. The results show that the proposed method leads to better symbolic regression results when compared with state-of-the-art methods that can select features directly from incomplete data. Further results show that our approach not only leads to better symbolic regression accuracy but also selects a smaller number of relevant features, and consequently improves both the effectiveness and the efficiency of the learning process.
We introduce a population-based approach to solving parameterized graph problems for which the goal is to identify a small set of vertices subject to a feasibility criterion. The idea is to evolve a population of individuals where each individual corresponds to an optimal solution to a subgraph of the original problem. The crossover operation then combines both solutions and subgraphs with the hope to generate an optimal solution for a slightly larger graph. In order to correctly combine solutions and subgraphs, we propose a new crossover operator called generalized allelic crossover which generalizes uniform crossover by associating a probability at each locus depending on the combined alleles of the parents. We prove for graphs with n vertices and m edges, the approach solves the k-vertex cover problem in expected time O( 4^k m + m^4 log n ) using a simple RLS-style mutation. This bound can be improved to O( 4^k m + m^2 n k log n) by using standard mutation constrained to the vertices of the graph.
