The Differential Evolution (DE) algorithm is one of the most successful evolutionary computation techniques. However, its structure is not trivially translatable in terms of mathematical transformations that describe its population dynamics. In this work, analytical expressions are developed for the probability of enhancement of individuals after each application of a mutation operator followed by a crossover operation, assuming a population distributed radially around the optimum for the sphere objective function, considering the DE/rand/1/bin and the DE/rand/1/exp algorithm versions. These expressions are validated by numerical experiments. Considering quadratic functions given by f(x)=xTDTDx and populations distributed according to the linear transformation D-1 of a radially distributed population, it is also shown that the expressions still hold in the cases when f(x) is separable (D is diagonal) and when D is any nonsingular matrix and the crossover rate is Cr=1.0. The expressions are employed for the analysis of DE population dynamics. The analysis is extended to more complex situations, reaching rather precise predictions of the effect of problem dimension and of the choice of algorithm parameters.

Comparing combinatorial optimization problems is a difficult task. They are defined using different criteria and terms: weights, flows, distances, etc. In spite of this apparent discrepancy, on many occasions, they tend to produce problem instances with similar properties. One avenue to compare different problems is to project them onto the same space, in order to have homogeneous representations. Expressing the problems in a unified framework could also lead to the discovery of theoretical properties or the design of new algorithms. This article proposes the use of the Fourier transform over the symmetric group as the tool to project different permutation-based combinatorial optimization problems onto the same space. Based on a previous study (Kondor, 2010), which characterized the Fourier coefficients of the quadratic assignment problem, we describe the Fourier coefficients of three other well-known problems: the symmetric and nonsymmetric traveling salesperson problem and the linear ordering problem. This transformation allows us to gain a better understanding of the intersection between the problems, as well as to bound their intrinsic dimension.

The traveling tournament problem is a well-known sports league scheduling problem famous for its practical hardness. Given an even number of teams with symmetric distances between their venues, a double round-robin tournament has to be scheduled minimizing the total travel distances over all teams. We consider the most common constrained variant without repeaters and a streak limit of three, for which we study a beam search approach based on a state-space formulation guided by heuristics derived from different lower bound variants. We solve the arising capacitated vehicle routing subproblems either exactly for small- to medium-sized instances up to 18 teams or heuristically also for larger instances up to 24 teams. In a randomized variant of the search, we employ random team ordering and add small amounts of Gaussian noise to the nodes' guidance for diversification when multiple runs are performed. This allows for a simple yet effective parallelization of the beam search. A final comparison is done on the NL, CIRC, NFL, and GALAXY benchmark instances with 12 to 24 teams, for which we report a mean gap difference to the best known feasible solutions of 1.2% and five new best feasible solutions.

Quality diversity algorithms can be used to efficiently create a diverse set of solutions to inform engineers' intuition. But quality diversity is not efficient in very expensive problems, needing hundreds of thousands of evaluations. Even with the assistance of surrogate models, quality diversity needs hundreds or even thousands of evaluations, which can make its use infeasible. In this study, we try to tackle this problem by using a pre-optimization strategy on a lower-dimensional optimization problem and then map the solutions to a higher-dimensional case. For a use case to design buildings that minimize wind nuisance, we show that we can predict flow features around 3D buildings from 2D flow features around building footprints. For a diverse set of building designs, by sampling the space of 2D footprints with a quality diversity algorithm, a predictive model can be trained that is more accurate than when trained on a set of footprints that were selected with a space-filling algorithm like the Sobol sequence. Simulating only 16 buildings in 3D, a set of 1,024 building designs with low predicted wind nuisance is created. We show that we can produce better machine learning models by producing training data with quality diversity instead of using common sampling techniques. The method can bootstrap generative design in a computationally expensive 3D domain and allow engineers to sweep the design space, understanding wind nuisance in early design phases.

Recently, Rowe and Aishwaryaprajna (2019) introduced a simple majority vote technique that efficiently solves Jump with large gaps, OneMax with large noise, and any monotone function with a polynomial-size image. In this paper, we identify a pathological condition for this algorithm: the presence of spin-flip symmetry in the problem instance. Spin-flip symmetry is the invariance of a pseudo-Boolean function to complementation. Many important combinatorial optimization problems admit objective functions that exhibit this pathology, such as graph problems, Ising models, and variants of propositional satisfiability. We prove that no population size exists that allows the majority vote technique to solve spin-flip symmetric functions of unitation with reasonable probability. To remedy this, we introduce a symmetry-breaking technique that allows the majority vote algorithm to overcome this issue for many landscapes. This technique requires only a minor modification to the original majority vote algorithm to force it to sample strings in {0,1}n from a dimension n-1 hyperplane. We prove a sufficient condition for a spin-flip symmetric function to possess in order for the symmetry-breaking voting algorithm to succeed, and prove its efficiency on generalized TwoMax, a spin-flip symmetric variant of Jump, and families of constructed 3-NAE-SAT and 2-XOR-SAT formulas. We also prove that the algorithm fails on the one-dimensional Ising model, and suggest different techniques for overcoming this. Finally, we present empirical results that explore the tightness of the runtime bounds and the performance of the technique on randomized satisfiability variants.

We present an empirical study of a range of evolutionary algorithms applied to various noisy combinatorial optimisation problems. There are three sets of experiments. The first looks at several toy problems, such as OneMax and other linear problems. We find that UMDA and the Paired-Crossover Evolutionary Algorithm (PCEA) are the only ones able to cope robustly with noise, within a reasonable fixed time budget. In the second stage, UMDA and PCEA are then tested on more complex noisy problems: SubsetSum, Knapsack, and SetCover. Both perform well under increasing levels of noise, with UMDA being the better of the two. In the third stage, we consider two noisy multiobjective problems (CountingOnesCountingZeros and a multiobjective formulation of SetCover). We compare several adaptations of UMDA for multiobjective problems with the Simple Evolutionary Multiobjective Optimiser (SEMO) and NSGA-II. We conclude that UMDA, and its variants, can be highly effective on a variety of noisy combinatorial optimisation, outperforming many other evolutionary algorithms.

In this paper we compare Bayesian Optimization, Differential Evolution, and an Evolution Strategy, employed as a gait learning algorithm in modular robots. The motivational scenario is the joint evolution of morphologies and controllers, where 'newborn' robots also undergo a learning process to optimize their inherited controllers (without changing their bodies). This context raises the question: How do gait learning algorithms compare when applied to various morphologies that are not known in advance (thus need to be treated without priors)? To answer this question, we use a test suite of twenty different robot morphologies to evaluate our gait learners and compare their efficiency, efficacy, and sensitivity to morphological differences. The results indicate that Bayesian Optimization and Differential Evolution deliver the same solution quality (walking speed for the robot) with fewer evaluations than the Evolution Strategy. Furthermore, the Evolution Strategy is more sensitive for morphological differences (its efficacy varies more between different morphologies) and is more subject to luck (repeated runs on the same morphology show greater variance in the outcomes).

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's 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.

Reproducibility is important for having confidence in evolutionary machine learning algorithms. Although the focus of reproducibility is usually to recreate an aggregate prediction error score using fixed random seeds, this is not sufficient. Firstly, multiple runs of an algorithm, without a fixed random seed, should ideally return statistically equivalent results. Secondly, it should be confirmed whether the expected behaviour of an algorithm matches its actual behaviour, in terms of how an algorithm targets a reduction in prediction error. Confirming the behaviour of an algorithm is not possible when using a total error aggregate score. Using an error decomposition framework as a methodology for improving the reproducibility of results in evolutionary computation addresses both of these factors. By estimating decomposed error using multiple runs of an algorithm and multiple training sets, the framework provides a greater degree of certainty about the prediction error. Also, decomposing error into bias, variance due to the algorithm (internal variance) and variance due to the training data (external variance) more fully characterises evolutionary algorithms. This allows the behaviour of an algorithm to be confirmed. Applying the framework to a number of evolutionary algorithms shows that their expected behaviour can be different to their actual behaviour. Identifying a behaviour mismatch is important in terms of understanding how to further refine an algorithm as well as how to effectively apply an algorithm to a problem.

In the traffic light scheduling problem, the evaluation of candidate solutions requires the simulation of a process under various (traffic) scenarios. Thus, good solutions should not only achieve good objective function values, but they must be robust (low variance) across all different scenarios. Previous work has shown that combining IRACE with evolutionary operators is effective for this task due to the power of evolutionary operators in numerical optimization. In this article, we further explore the hybridization of evolutionary operators and the elitist iterated racing of IRACE for the simulation-optimization of traffic light programs. We review previous works from the literature to find the evolutionary operators performing the best when facing this problem to propose new hybrid algorithms. We evaluate our approach over a realistic case study derived from the traffic network of Málaga (Spain) with 275 traffic lights that should be scheduled optimally. The experimental analysis reveals that the hybrid algorithm comprising IRACE plus differential evolution offers statistically better results than the other algorithms when the budget of simulations is low. In contrast, IRACE performs better than the hybrids for a high simulations budget, although the optimization time is much longer.

Recently a mechanism called stagnation detection was proposed that automatically adjusts the mutation rate of evolutionary algorithms when they encounter local optima. The so-called SD-(1+1) EA introduced by Rajabi and Witt (2022) adds stagnation detection to the classical (1+1) EA with standard bit mutation. This algorithm flips each bit independently with some mutation rate, and stagnation detection raises the rate when the algorithm is likely to have encountered a local optimum. In this article, we investigate stagnation detection in the context of the k-bit flip operator of randomized local search that flips k bits chosen uniformly at random and let stagnation detection adjust the parameter k. We obtain improved runtime results compared with the SD-(1+1) EA amounting to a speedup of at least (1-o(1))2πm, where m is the so-called gap size, that is, the distance to the next improvement. Moreover, we propose additional schemes that prevent infinite optimization times even if the algorithm misses a working choice of k due to unlucky events. Finally, we present an example where standard bit mutation still outperforms the k-bit flip operator with stagnation detection.

Model management is an essential component in data-driven surrogate-assisted evolutionary optimization. In model management, the solutions with a large degree of uncertainty in approximation play an important role. They can strengthen the exploration ability of algorithms and improve the accuracy of surrogates. However, there is no theoretical method to measure the uncertainty of prediction of Non-Gaussian process surrogates. To address this issue, this article proposes a method to measure the uncertainty. In this method, a stationary random field with a known zero mean is used to measure the uncertainty of prediction of Non-Gaussian process surrogates. Based on experimental analyses, this method is able to measure the uncertainty of prediction of Non-Gaussian process surrogates. The method's effectiveness is demonstrated on a set of benchmark problems in single surrogate and ensemble surrogates cases.

We contribute to the efficient approximation of the Pareto-set for the classical NP-hard multi-objective minimum spanning tree problem (moMST) adopting evolutionary computation. More precisely, by building upon preliminary work, we analyse the neighborhood structure of Pareto-optimal spanning trees and design several highly biased sub-graph-based mutation operators founded on the gained insights. In a nutshell, these operators replace (un)connected sub-trees of candidate solutions with locally optimal sub-trees. The latter (biased) step is realized by applying Kruskal's single-objective MST algorithm to a weighted sum scalarization of a sub-graph. We prove runtime complexity results for the introduced operators and investigate the desirable Pareto-beneficial property. This property states that mutants cannot be dominated by their parent. Moreover, we perform an extensive experimental benchmark study to showcase the operator's practical suitability. Our results confirm that the subgraph based operators beat baseline algorithms from the literature even with severely restricted computational budget in terms of function evaluations on four different classes of complete graphs with different shapes of the Pareto-front.

When it comes to solving optimization problems with evolutionary algorithms (EAs) in a reliable and scalable manner, detecting and exploiting linkage information, i.e., dependencies between variables, can be key. In this article, we present the latest version of, and propose substantial enhancements to, the Gene-pool Optimal Mixing Evoutionary Algorithm (GOMEA): an EA explicitly designed to estimate and exploit linkage information. We begin by performing a largescale search over several GOMEA design choices to understand what matters most and obtain a generally best-performing version of the algorithm. Next, we introduce a novel version of GOMEA, called CGOMEA, where linkage-based variation is further improved by filtering solution mating based on conditional dependencies. We compare our latest version of GOMEA, the newly introduced CGOMEA, and another contending linkage-aware EA, DSMGA-II, in an extensive experimental evaluation, involving a benchmark set of 9 black-box problems that can only be solved efficiently if their inherent dependency structure is unveiled and exploited. Finally, in an attempt to make EAs more usable and resilient to parameter choices, we investigate the performance of different automatic population management schemes for GOMEA and CGOMEA, de facto making the EAs parameterless. Our results show that GOMEA and CGOMEA significantly outperform the original GOMEA and DSMGA-II on most problems, setting a new state of the art for the field.

Territorial Differential Meta-Evolution (TDME) is an efficient, versatile, and reliable algorithm for seeking all the global or desirable local optima of a multivariable function. It employs a progressive niching mechanism to optimize even challenging, highdimensional functions with multiple global optima and misleading local optima. This article introduces TDME and uses standard and novel benchmark problems to quantify its advantages over HillVallEA, which is the best-performing algorithm on the standard benchmark suite that has been used by all major multimodal optimization competitions since 2013. TDME matches HillVallEA on that benchmark suite and categorically outperforms it on a more comprehensive suite that better reflects the potential diversity of optimization problems. TDME achieves that performance without any problem-specific parameter tuning.

Exposing an Evolutionary Algorithm that is used to evolve robot controllers to variable conditions is necessary to obtain solutions which are robust and can cross the reality gap. However, we do not yet have methods for analyzing and understanding the impact of the varying morphological conditions which impact the evolutionary process, and therefore for choosing suitable variation ranges. By morphological conditions, we refer to the starting state of the robot, and to variations in its sensor readings during operation due to noise. In this article, we introduce a method that permits us to measure the impact of these morphological variations and we analyze the relation between the amplitude of variations, the modality with which they are introduced, and the performance and robustness of evolving agents. Our results demonstrate that (i) the evolutionary algorithm can tolerate morphological variations which have a very high impact, (ii) variations affecting the actions of the agent are tolerated much better than variations affecting the initial state of the agent or of the environment, and (iii) improving the accuracy of the fitness measure through multiple evaluations is not always useful. Moreover, our results show that morphological variations permit generating solutions which perform better both in varying and non-varying conditions.

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.

We reflect on 30 years of the journal Evolutionary Computation. Taking the papers published in the first volume in 1993 as a springboard, as the founding and current Editors-in-Chief, we comment on the beginnings of the field, evaluate the extent to which the field has both grown and itself evolved, and provide our own perpectives on where the future lies.

This paper presents a personal account of the author's 35 years "adventure" with Evolutionary Computation-from the first encounter in 1988 and many years of academic research through to working full-time in business-successfully implementing evolutionary algorithms for some of the world's largest corporations. The paper concludes with some observations and insights.

On the occasion of the 30-year anniversary of the Evolutionary Computation journal, I was invited by Professor Hart to offer some reflections on the article on evolving behaviors in the iterated prisoner's dilemma that I contributed to its first issue in 1993. It's an honor to do so. I would like to thank Professor Ken De Jong, the journal's first editor-in-chief, for his vision in creating the journal, and the editors who have followed and maintained that vision. This article contains some personal reflections on the topic and the field as a whole.