Structural bias (SB) refers to systematic preferences of an optimisation algorithm for particular regions of the search space that arise independently of the objective function. While SB has been studied extensively in single-objective optimisation, its role in multi-objective optimisation remains largely unexplored. This is problematic, as dominance relations, diversity preservation and Pareto-based selection mechanisms may introduce or amplify structural effects.

The advent of Large Language Models (LLMs) has opened new frontiers in automated algorithm design, giving rise to numerous powerful methods. However, these approaches retain critical limitations: they require extensive evaluation of the target problem to guide the search process, making them impractical for real-world optimization tasks, where each evaluation consumes substantial computational resources. This research proposes an innovative and efficient framework that decouples algorithm discovery from high-cost evaluation. Our core innovation lies in combining a Genetic Programming (GP) function generator with an LLM-driven evolutionary algorithm designer. The evolutionary direction of the GP-based function generator is guided by the similarity between the landscape characteristics of generated proxy functions and those of real-world problems, ensuring that algorithms discovered via proxy functions exhibit comparable performance on real-world problems. Our method enables deep exploration of the algorithmic space before final validation while avoiding costly real-world evaluations. We validate the framework's efficacy across multiple real-world problems, demonstrating its ability to discover high-performance algorithms while substantially reducing expensive evaluations. This approach shows a path to apply LLM-based automated algorithm design to computationally intensive real-world optimization challenges.

The mutation process in evolution strategies has been interlinked with the normal distribution since its inception. Many lines of reasoning have been given for this strong dependency, ranging from maximum entropy arguments to the need for isotropy. However, some theoretical results suggest that other distributions might lead to similar local convergence properties. This paper empirically shows that a wide range of evolutionary strategies, from the (1+1)-ES to CMA-ES, show comparable optimization performance when using a mutation distribution other than the standard Gaussian. Replacing it with, e.g., uniformly distributed mutations, does not deteriorate the performance of ES, when using the default adaptation mechanism for the strategy parameters. We observe that these results hold not only for the sphere model but also for a wider range of benchmark problems.

Large Language Models (LLMs) have demonstrated great promise in generating code, especially when used inside an evolutionary computation framework to iteratively optimize the generated algorithms. However, in some cases they fail to generate competitive algorithms or the code optimization stalls, and we are left with no recourse because of a lack of understanding of the generation process and generated codes. We present a novel approach to mitigate this problem by enabling users to analyze the generated codes inside the evolutionary process and how they evolve over repeated prompting of the LLM. We show results for three benchmark problem classes and demonstrate novel insights. In particular, LLMs tend to generate more complex code with repeated prompting, but additional complexity can hurt algorithmic performance in some cases. Different LLMs have different coding "styles" and generated code tends to be dissimilar to other LLMs. These two findings suggest that using different LLMs inside the code evolution frameworks might produce higher performing code than using only one LLM.

Benchmarking heuristic algorithms is vital to understand under which conditions and on what kind of problems certain algorithms perform well. In most current research into heuristic optimization algorithms, only a very limited number of scenarios, algorithm configurations and hyper-parameter settings are explored, leading to incomplete and often biased insights and results. This article presents a novel approach that we call explainable benchmarking. We introduce the IOHxplainer software library, for systematic analysing the performance of various optimization algorithms and the impact of their different components and hyperparameters. We showcase the methodology in the context of two modular optimization implementations. Through this library, we examine the impact of different algorithmic components and configurations, offering insights into their performance across diverse scenarios. We provide a systematic method for evaluating and interpreting the behaviour and efficiency of iterative optimization heuristics in a more transparent and comprehensible manner, aiming to improve future benchmarking and algorithm design practices.

Symbolic regression (SR) poses a significant challenge for randomized search heuristics due to its reliance on the synthesis of expressions for input-output mappings. Although traditional genetic programming (GP) algorithms have achieved success in various domains, they exhibit limited performance when tree-based representations are used for SR. To address these limitations, we introduce a novel SR approach called Fourier Tree Growing (FTG) that draws insights from functional analysis. This new perspective enables us to perform optimization directly in a different space, thus avoiding intricate symbolic expressions. Our proposed algorithm exhibits significant performance improvements over traditional GP methods on a range of classical one-dimensional benchmarking problems. To identify and explain the limiting factors of GP and FTG, we perform experiments on a large-scale polynomials benchmark with high-order polynomials up to degree 100. To the best of the authors' knowledge, this work represents the pioneering application of functional analysis in addressing SR problems. The superior performance of the proposed algorithm and insights into the limitations of GP open the way for further advancing GP for SR and related areas of explainable machine learning.

The number of proposed iterative optimization heuristics is growing steadily, and with this growth, there have been many points of discussion within the wider community. One particular criticism that is raised towards many new algorithms is their focus on metaphors used to present the method, rather than emphasizing their potential algorithmic contributions. Several studies into popular metaphor-based algorithms have highlighted these problems, even showcasing algorithms that are functionally equivalent to older existing methods. Unfortunately, this detailed approach is not scalable to the whole set of metaphor-based algorithms. Because of this, we investigate ways in which benchmarking can shed light on these algorithms. To this end, we run a set of 294 algorithm implementations on the BBOB function suite. We investigate how the choice of the budget, the performance measure, or other aspects of experimental design impact the comparison of these algorithms. Our results emphasize why benchmarking is a key step in expanding our understanding of the algorithm space, and what challenges still need to be overcome to fully gauge the potential improvements to the state-of-the-art hiding behind the metaphors.

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is one of the most successful examples of a derandomized evolution strategy. However, it still relies on randomly sampling offspring, which can be done via a uniform distribution and subsequently transforming into the required Gaussian. Previous work has shown that replacing this uniform sampling with a low-discrepancy sampler, such as Halton or Sobol sequences, can improve performance over a wide set of problems. We show that iterating through small, fixed sets of low-discrepancy points can still perform better than the default uniform distribution. Moreover, using only 128 points throughout the search is sufficient to closely approximate the empirical performance of using the complete pseudorandom sequence up to dimensionality 40 on the BBOB benchmark. For lower dimensionalities (below 10), we find that using as little as 32 unique low discrepancy points performs similar or better than uniform sampling. In 2D, for which w e have highly optimized low discrepancy samples available, we demonstrate that using these points yields the highest empirical performance and requires only 16 samples to improve over uniform sampling. Overall, we establish a clear relation between the L2 discrepancy of the used point set and the empirical performance of the CMA-ES.

When benchmarking optimization heuristics, we need to take care to avoid an algorithm exploiting biases in the construction of the used problems. One way in which this might be done is by providing different versions of each problem but with transformations applied to ensure the algorithms are equipped with mechanisms for successfully tackling a range of problems. In this paper, we investigate several of these problem transformations and show how they influence the low-level landscape features of problems from the Congress on Evolutionary Computation 2022 benchmark suite. Our results highlight that even relatively small transformations can significantly alter the measured landscape features. This poses a wider question of what properties we want to preserve when creating problem transformations, and how to measure them fairly.

Abstract To guide the design of better iterative optimisation heuristics, it is imperative to understand how inherent structural biases within algorithm components affect the performance on a wide variety of search landscapes. This study explores the impact of structural bias in the modular Covariance Matrix Adaptation Evolution Strategy (modCMA), focusing on the roles of various modulars within the algorithm. Through an extensive investigation involving $$435\,456$$ 435 456 configurations of modCMA, we identified key modules that significantly influence structural bias of various classes. Our analysis utilized the Deep-BIAS toolbox for structural bias detection and classification, complemented by SHAP analysis for quantifying module contributions. The performance of these configurations was tested on a sequence of affine-recombined functions, maintaining fixed optimum locations while gradually varying the landscape features. Our results demonstrate an interplay between module-induced structural bias and algorithm performance across different landscape characteristics.

Time-series anomaly detection plays an important role in engineering processes, like development, manufacturing and other operations involving dynamic systems. These processes can greatly benefit from advances in the field, as state-of-the-art approaches may aid in cases involving, for example, highly dimensional data. To provide the reader with understanding of the terminology, this survey introduces a novel taxonomy where a distinction between online and offline, and training and inference is made. Additionally, it presents the most popular data sets and evaluation metrics used in the literature, as well as a detailed analysis. Furthermore, this survey provides an extensive overview of the state-of-the-art model-based online semi- and unsupervised anomaly detection approaches for multivariate time-series data, categorising them into different model families and other properties. The biggest research challenge revolves around benchmarking, as currently there is no reliable way to compare different approaches against one another. This problem is two-fold: on the one hand, public data sets suffers from at least one fundamental flaw, while on the other hand, there is a lack of intuitive and representative evaluation metrics in the field. Moreover, the way most publications choose a detection threshold disregards real-world conditions, which hinders the application in the real world. To allow for tangible advances in the field, these issues must be addressed in future work.

We argue that results produced by a heuristic optimisation algorithm cannot be considered reproducible unless the algorithm fully specifies what should be done with solutions generated outside the domain, even in the case of simple bound constraints. Currently, in the field of heuristic optimisation, such specification is rarely mentioned or investigated due to the assumed triviality or insignificance of this question. Here, we demonstrate that, at least in algorithms based on Differential Evolution, this choice induces notably different behaviours in terms of performance, disruptiveness, and population diversity. This is shown theoretically (where possible) for standard Differential Evolution in the absence of selection pressure and experimentally for the standard and state-of-the-art Differential Evolution variants, on a special test function and the BBOB benchmarking suite, respectively. Moreover, we demonstrate that the importance of this choice quickly grows with problem dimensionality. Differential Evolution is not at all special in this regard-there is no reason to presume that other heuristic optimisers are not equally affected by the aforementioned algorithmic choice. Thus, we urge the heuristic optimisation community to formalise and adopt the idea of a new algorithmic component in heuristic optimisers, which we refer to as the strategy of dealing with infeasible solutions. This component needs to be consistently: (a) specified in algorithmic descriptions to guarantee reproducibility of results, (b) studied to better understand its impact on an algorithm's performance in a wider sense (i.e., convergence time, robustness, etc.), and (c) included in the (automatic) design of algorithms. All of these should be done even for problems with bound constraints.

The objective value of an ill-conditioned function may significantly change with a minor shift of the argument in the search space. Ill-conditioned functions do not have at all or exhibit very few hints towards better solutions and, thus, they are usually difficult to optimize with randomized search heuristics. However, problems that emerge in practical applications are likely to be formulated as ill-conditioned functions, as often Euclidean metric is used to measure distance in the search space. At the same time, it may be possible to use domain-specific knowledge to define such a metric in the search space so that the function stops being ill-conditioned. We consider finite search spaces and propose two mutation operators that leverage such metric to optimize the function more efficiently. The first operator assumes prior knowledge about the distance, the second operator uses the distance as a black box. Those operators apply an estimation of distribution algorithm to find the best mutant according to the defined function, which employs the given metric. For pseudo-Boolean and integer optimization problems, we experimentally show that both mutation operators speed up the search on most of the functions when applied in considered evolutionary algorithms and random local search. Moreover, those operators can be applied in any randomized search heuristic which uses perturbations. However, our mutation operators increase wall-clock time and so are helpful in practice when distance is (much) cheaper to compute than the real objective function.

The domain of an optimization problem is seen as one of its most important characteristics. In particular, the distinction between continuous and discrete optimization is rather impactful. Based on this, the optimizing algorithm, analyzing method, and more are specified. However, in practice, no problem is ever truly continuous. Whether this is caused by computing limits or more tangible properties of the problem, most variables have a finite resolution.

New contributions in the field of iterative optimisation heuristics are often made in an iterative manner. Novel algorithmic ideas are not proposed in isolation, but usually as extensions of a preexisting algorithm. Although these contributions are often compared to the base algorithm, it is challenging to make fair comparisons between larger sets of algorithm variants. This happens because even small changes in the experimental setup, parameter settings, or implementation details can cause results to become incomparable. Modular algorithms offer a way to overcome these challenges. By implementing the algorithmic modifications into a common framework, many algorithm variants can be compared, while ensuring that implementation details match in all versions.

A clear need for automatic anomaly detection applied to automotive testing has emerged as more and more attention is paid to the data recorded and manual evaluation by humans reaches its capacity. Such real-world data is massive, diverse, multivariate and temporal in nature, therefore requiring modelling of the testee behaviour. We propose a variational autoencoder with multi-head attention (MA-VAE), which, when trained on unlabelled data, not only provides very few false positives but also manages to detect the majority of the anomalies presented. In addition to that, the approach offers a novel way to avoid the bypass phenomenon, an undesirable behaviour investigated in literature. Lastly, the approach also introduces a new method to remap individual windows to a continuous time series. The results are presented in the context of a real-world in-dustrial data set and several experiments are undertaken to further investigate certain aspects of the proposed model. When configured pro perly, it is 9% of the time wrong when an anomaly is flagged and discovers 67% of the anomalies present. Also, MA-VAE has the potential to perform well with only a fraction of the training and validation subset, however, to extract it, a more sophisticated threshold estimation method is required.

The algorithm selection problem is of paramount importance in achieving high-quality results while minimizing computational effort, especially when dealing with expensive black-box optimization problems. In this paper, we address this challenge by using randomly generated artificial functions that mimic the landscape characteristics of the original problem while being inexpensive to evaluate. The similarity between the artificial function and the original problem is quantified using Exploratory Landscape Analysis. We demonstrate a significant performance improvement on five real-world vehicle dynamics problems by transferring the parameters of the Covariance Matrix Adaptation Evolution Strategy tuned to these artificial functions. We provide a complete set of simulated values of braking distance for fully enumerated 2D design spaces of all five real-world optimization problems. So, replication of our results and benchmarking directly on the real-world problems is possible. Beyond the scope of this paper, this data can be used as a benchmarking set for multi-objective optimization with up to five objectives.

The performance of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is significantly affected by the selection of the specific CMA-ES variant and the parameter values used. Furthermore, optimal CMA-ES parameter configurations vary across different problem landscapes, making the task of tuning CMA-ES to a specific optimization problem a challenging mixed-integer optimization problem. In recent years, several advanced algorithms have been developed to address this problem, including the Sequential Model-based Algorithm Configuration (SMAC) and the Tree-structured Parzen Estimator (TPE). In this study, we propose a novel approach for tuning CMA-ES by leveraging CMA-ES itself. Therefore, we combine the modular CMA-ES implementation with the margin extension to handle mixed-integer optimization problems. We show that CMA-ES can not only compete with SMAC and TPE but also outperform them in terms of wall clock time.

Thirty years, 1993-2023, is a huge time frame in science. We address some major developments in the field of evolutionary algorithms, with applications in parameter optimization, over these 30 years. These include the covariance matrix adaptation evolution strategy and some fast-growing fields such as multimodal optimization, surrogate-assisted optimization, multiobjective optimization, and automated algorithm design. Moreover, we also discuss particle swarm optimization and differential evolution, which did not exist 30 years ago, either. One of the key arguments made in the paper is that we need fewer algorithms, not more, which, however, is the current trend through continuously claiming paradigms from nature that are suggested to be useful as new optimization algorithms. Moreover, we argue that we need proper benchmarking procedures to sort out whether a newly proposed algorithm is useful or not. We also briefly discuss automated algorithm design approaches, including configurable algorithm design frameworks, as the proposed next step toward designing optimization algorithms automatically, rather than by hand.

Benchmarking heuristic algorithms is vital to understand under which conditions and on what kind of problems certain algorithms perform well. Most benchmarks are performance based, to test algorithm performance under a wide set of conditions. There is also resource- and behavior-based benchmarks to test the resource consumption and the behavior of algorithms. In this article, we propose a novel behavior-based benchmark toolbox: BIAS (Bias in algorithms, structural). This toolbox can detect structural bias (SB) per dimension and across dimension-based on 39 statistical tests. Moreover, it predicts the type of SB using a random forest model. BIAS can be used to better understand and improve existing algorithms (removing bias) as well as to test novel algorithms for SB in an early phase of development. Experiments with a large set of generated SB scenarios show that BIAS was successful in identifying bias. In addition, we also provide the results of BIAS on 432 existing state-of-the-art optimization algorithms showing that different kinds of SB are present in these algorithms, mostly toward the center of the objective space or showing discretization behavior. The proposed toolbox is made available open-source and recommendations are provided for the sample size and hyper-parameters to be used when applying the toolbox on other algorithms.

The imbalanced classification problem is very relevant in both academic and industrial applications. The task of finding the best machine learning model to use for a specific imbalanced dataset is complicated due to a large number of existing algorithms, each with its own hyperparameters. The Combined Algorithm Selection and Hyperparameter optimization (CASH) has been introduced to tackle both aspects at the same time. However, CASH has not been studied in detail in the class imbalance domain, where the best combination of resampling technique and classification algorithm is searched for, together with their optimized hyperparameters. Thus, we target the CASH problem for imbalanced classification. We experiment with a search space of 5 classification algorithms, 21 resampling approaches and 64 relevant hyperparameters in total. Moreover, we investigate performance of 2 well-known optimization approaches: Random search and Tree Parzen Estimators approach which is a kind of Bayesian optimization. For comparison, we also perform grid search on all combinations of resampling techniques and classification algorithms with their default hyperparameters. Our experimental results show that a Bayesian optimization approach outperforms the other approaches for CASH in this application domain.

This paper investigates whether optimisation methods with the population made up of one solution can suffer from structural bias just like their multisolution variants. Following recent results highlighting the importance of choice of strategy for handling solutions generated outside the domain, a selection of single solution methods are considered in conjunction with several such strategies. Obtained results are tested for the presence of structural bias by means of a traditional approach from literature and a newly proposed here statistical approach. These two tests are demonstrated to be not fully consistent. All tested methods are found to be structurally biased with at least one of the tested strategies. Confirming results for multisolution methods, it is such strategy that is shown to control the emergence of structural bias in single solution methods. Some of the tested methods exhibit a kind of structural bias that has not been observed before.

This paper proposes a scheme for the hybridisation of an evolution strategy framework and periodically scheduled Nelder-Mead algorithm. This relatively simple hybridisation scheme turns out to be efficient for the optimisation problems in higher dimensions. The efficiency of the proposed method is tested for a complex engineering problem, namely an inverse problem of chemical kinetics. An extensive parameter analysis and tuning are presented. Numerical results show the superiority of the proposed methods in comparison with some popular metaheuristics and some tailored algorithms presented in the literature for solving the problem under investigation.

This paper proposes the fitness diversity based adaptive memetic algorithm (FIDAMA) for solving the problem of the inverse type consisting of retrieving chemical kinetics reaction rate coefficients in the generalised Arrhenius form based on the observed concentrations in a given range of temperatures of a limited set of species which describe the reaction mechanism. FIDAMA consists of the evolutionary framework and three local searchers adaptively governed by a novel fitness diversity based measure. Moreover, a certain simplification of the decision space was carried out without any deterioration in the result obtained. The numerical results preseted show the superiority of FIDAMA compared to the other published computational intelligence methods.