The most common representation in evolutionary computation are bit strings. This is ideal to model binary decision variables, but less useful for variables taking more values. With very little theoretical work existing on how to use evolutionary algorithms for such optimization problems, we study the run time of simple evolutionary algorithms on some OneMax-like functions defined over $\Omega = \{0, 1, \dots, r-1\}^n$. More precisely, we regard a variety of problem classes requesting the component-wise minimization of the distance to an unknown target vector $z \in \Omega$. For such problems we see a crucial difference in how we extend the standard-bit mutation operator to these multi-valued domains. While it is natural to select each position of the solution vector to be changed independently with probability $1/n$, there are various ways to then change such a position. If we change each selected position to a random value different from the original one, we obtain an expected run time of $\Theta(nr \log n)$. If we change each selected position by either $+1$ or $-1$ (random choice), the optimization time reduces to $\Theta(nr + n\log n)$. If we use a random mutation strength $i \in \{0,1,\ldots,r-1\}^n$ with probability inversely proportional to $i$ and change the selected position by either $+i$ or $-i$ (random choice), then the optimization time becomes $\Theta(n \log(r)(\log(n)+\log(r)))$, bringing down the dependence on $r$ from linear to polylogarithmic. One of our results depends on a new variant of the lower bounding multiplicative drift theorem.
One of the common artificial intelligence applications in electronic games consists of making an artificial agent learn how to execute some determined task successfully in a game environment. One way to perform this task is through machine learning algorithms capable of learning the sequence of actions required to win in a given game environment. There are several supervised learning techniques able to learn the correct answer for a problem through examples. However, when learning how to play electronic games, the correct answer might only be known by the end of the game, after all the actions were already taken. Thus, not being possible to measure the accuracy of each individual action to be taken at each time step. A way for dealing with this problem is through Neuroevolution, a method which trains Artificial Neural Networks using evolutionary algorithms. In this article, we introduce a framework for testing optimization algorithms with artificial agent controllers in electronic games, called EvoMan, which is inspired in the action-platformer game Mega Man II. The environment can be configured to run in different experiment modes, as single evolution, coevolution and others. To demonstrate some challenges regarding the proposed platform, as initial experiments we applied Neuroevolution using Genetic Algorithms and the NEAT algorithm, in the context of competitively coevolving two distinct agents in this game.
Differential Evolution (DE) proved to be one of the most successful evolutionary algorithms for global optimization purposes in continuous problems. The core operator in DE is mutation which can provide the algorithm with both exploration and exploitation. In this article, a new notation for DE is proposed which has a formula that can be utilized for generating and extracting novel mutations and by applying this new notation, four novel mutations are proposed. More importantly, by combining these novel trial vector generation strategies and four other well-known ones, we proposed Generalized Mutation Differential Evolution (GMDE) that takes advantage of two mutation pools that have both explorative and exploitative strategies inside them. Results and experimental analysis are performed on CEC2005 benchmarks and the results stated that GMDE is surprisingly competitive and significantly improved the performance of this algorithm. Finally, GMDE is also applied to parameters optimization, modification and improvement of a feature selection method for cancer classification purposes over gene expression microarray profiles.
Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-Howard isomorphism, proof assistants, software capable of interacting with human for constructing formal proofs, have been developed in the past several decades. Since proofs can be considered and expressed as programs, proof assistants simplify and verify a proof by computationally evaluating the program corresponding to the proof. Thanks to the transformation from logic to computation, it is now possible to generate or search for formal proofs directly in the realm of computation. Evolutionary algorithms, known to be flexible and versatile, have been successfully applied to handle a variety of scientific and engineering problems in numerous disciplines for also several decades. Examining the feasibility of establishing the link between evolutionary algorithms, as the program generator, and proof assistants, as the proof verifier, in order to automatically find formal proofs to a given logic sentence is the primary goal of this study. In the article, we describe in detail our first, ad-hoc attempt to fully automatically prove theorems as well as the preliminary results. Ten simple theorems from various branches of mathematics were proven, and most of these theorems cannot be proven by using the tactic auto alone in Coq, the adopted proof assistant. The implication and potential influence of this study are discussed, and the developed source code with the obtained experimental results are released as open source.
Evolutionary algorithms have been widely studied from a theoretical perspective. In particular, the area of runtime analysis has contributed significantly to a theoretical understanding and provided insights into the working behaviour of these algorithms. We study how these insights into evolutionary processes can be used for evolutionary art. We introduce the notion of evolutionary image transition which transfers a given starting image into a target image through an evolutionary process. Combining standard mutation effects known from the optimization of the classical benchmark function OneMax and different variants of random walks, we present ways of performing evolutionary image transition with different artistic effects.
There has been a variety of crossover operators proposed for Real-Coded Genetic Algorithms (RCGAs), which recombine values from the same location in pairs of strings. In this article we present a recombination operator for RC- GAs that selects the locations randomly in both parents, and compare it to mainstream crossover operators in a set of experiments on a range of standard multidimensional optimization problems and a clustering problem. We present two variants of the operator, either selecting both bits uniformly at random in the strings, or sampling the second bit from a normal distribution centered at the selected location in the first string. While the operator is biased towards exploitation of fitness space, the random selection of the second bit for swap- ping makes it slightly less exploitation-biased. Extensive statistical analysis using a non-parametric test shows the advantage of the new recombination operators on a range of test functions.
Differential Evolution (DE) is one of the most successful and powerful evolutionary algorithms for global optimization problem. The most important operator in this algorithm is mutation operator which parents are selected randomly to participate in it. Recently, numerous papers are tried to make this operator more intelligent by selection of parents for mutation intelligently. The intelligent selection for mutation vectors is performed by applying design space (also known as decision space) criterion or fitness space criterion, however, in both cases, half of valuable information of the problem space is disregarded. In this article, a Universal Differential Evolution (UDE) is proposed which takes advantage of both design and fitness spaces criteria for intelligent selection of mutation vectors. The experimental analysis on UDE are performed on CEC2005 benchmarks and the results stated that UDE significantly improved the performance of differential evolution in comparison with other methods that only use one criterion for intelligent selection.
In this article, we present the elitist particle filter based on evolutionary strategies (EPFES) as an efficient approach for nonlinear system identification. The EPFES is derived from the frequently-employed state-space model, where the relevant information of the nonlinear system is captured by an unknown state vector. Similar to classical particle filtering, the EPFES consists of a set of particles and respective weights which represent different realizations of the latent state vector and their likelihood of being the solution of the optimization problem. As main innovation, the EPFES includes an evolutionary elitist-particle selection which combines long-term information with instantaneous sampling from an approximated continuous posterior distribution. In this article, we propose two advancements of the previously-published elitist-particle selection process. Further, the EPFES is shown to be a generalization of the widely-used Gaussian particle filter and thus evaluated with respect to the latter for two completely different scenarios: First, we consider the so-called univariate nonstationary growth model with time-variant latent state variable, where the evolutionary selection of elitist particles is evaluated for non-recursively calculated particle weights. Second, the problem of nonlinear acoustic echo cancellation is addressed in a simulated scenario with speech as input signal: By using long-term fitness measures, we highlight the efficacy of the well-generalizing EPFES in estimating the nonlinear system even for large search spaces. Finally, we illustrate similarities between the EPFES and evolutionary algorithms to outline future improvements by fusing the achievements of both fields of research.
Evolutionary algorithms (EAs) are population-based general-purpose optimization algorithms, and have been successfully applied in various real-world optimization tasks. However, previous theoretical studies often employ EAs with only a parent or offspring population and focus on specific problems. Furthermore, they often only show upper bounds on the running time, while lower bounds are also necessary to get a complete understanding of an algorithm. In this paper, we analyze the running time of the ($\mu$+$\lambda$)-EA (a general population-based EA with mutation only) on the class of pseudo-Boolean functions with a unique global optimum. By applying the recently proposed switch analysis approach, we prove the lower bound $\Omega(n \ln n+ \mu + \lambda n\ln\ln n/ \ln n)$ for the first time. Particularly on the two widely-studied problems, OneMax and LeadingOnes, the derived lower bound discloses that the ($\mu$+$\lambda$)-EA will be strictly slower than the (1+1)-EA when the population size $\mu$ or $\lambda$ is above a moderate order. Our results imply that the increase of population size, while usually desired in practice, bears the risk of increasing the lower bound of the running time and thus should be carefully considered.
This paper aims to study how the population size affects the computation time of evolutionary algorithms in a rigorous way. The computation time of an evolutionary algorithm can be measured by either the expected number of generations (hitting time) or the expected number of fitness evaluations (running time) to find an optimal solution. Population scalability is the ratio of the expected hitting time between a benchmark algorithm and an algorithm using a larger population size. Average drift analysis is presented for comparing the expected hitting time of two algorithms and estimating lower and upper bounds on population scalability. Several intuitive beliefs are rigorously analysed. It is prove that (1) using a population sometimes increases rather than decreases the expected hitting time; (2) using a population cannot shorten the expected running time of any elitist evolutionary algorithm on unimodal functions in terms of the time-fitness landscape, but this is not true in terms of the distance-based fitness landscape; (3) using a population cannot always reduce the expected running time on fully-deceptive functions, which depends on the benchmark algorithm using elitist selection or random selection.
Evolutionary algorithms (EAs) have been well acknowledged as a promising paradigm for solving optimisation problems with multiple conflicting objectives in the sense that they are able to locate a set of diverse approximations of Pareto optimal solutions in a single run. EAs drive the search for approximated solutions through maintaining a diverse population of solutions and by recombining promising solutions selected from the population. Combining machine learning techniques has shown great potentials since the intrinsic structure of the Pareto optimal solutions of an multiobjective optimisation problem can be learned and used to guide for effective recombination. However, existing multiobjective EAs (MOEAs) based on structure learning spend too much computational resources on learning. To address this problem, we propose to use an online learning scheme. Based on the fact that offsprings along evolution are streamy, dependent and non-stationary (which implies that the intrinsic structure, if any, is temporal and scale-variant), an online agglomerative clustering algorithm is applied to adaptively discover the intrinsic structure of the Pareto optimal solution set; and to guide effective offspring recombination. Experimental results have shown significant improvement over five state-of-the-art MOEAs on a set of well-known benchmark problems with complicated Pareto sets and complex Pareto fronts.
The runtime of evolutionary algorithms (EAs) depends critically on their parameter settings, which are often problem-specific. Automated schemes for parameter tuning have been developed to alleviate the high costs of manual parameter tuning. Experimental results indicate that self-adaptation, where parameter settings are encoded in the genomes of individuals, can be effective in continuous optimisation. However, results in discrete optimisation have been less conclusive. Furthermore, a rigorous runtime analysis that explains how self-adaptation can lead to asymptotic speedups has been missing. This paper provides the first such analysis for discrete, population-based EAs. We apply level-based analysis to show how a self-adaptive EA is capable of fine-tuning its mutation rate, leading to exponential speedups over EAs using fixed mutation rates.
Real-world optimisation problems are often dynamic. Previously good solutions must be updated or replaced due to changes in objectives and constraints. It is often claimed that evolutionary algorithms are particularly suitable for dynamic optimisation because a large population can contain different solutions that may be useful in the future. However, rigorous theoretical demonstrations for how populations in dynamic optimisation can be essential are sparse and restricted to special cases. This paper provides theoretical explanations of how populations can be essential in evolutionary dynamic optimisation in a general and natural setting. We describe a natural class of dynamic optimisation problems where a sufficiently large population is necessary to keep track of moving optima reliably. We establish a relationship between the population-size and the probability that the algorithm loses track of the optimum.
Multi-population evolutionary algorithms are, by nature, highly complex and difficult to describe. Even two populations working in concert (or opposition) present a myriad of potential configurations that are often difficult to relate using text alone. Little effort has been made, however, to depict these kinds of systems, relying solely on the simple structural connections (related using ad hoc diagrams) between populations and often leaving out crucial details. In this paper, we propose a notation and accompanying formalism for consistently and powerfully depicting these structures and the relationships within them in an intuitive and consistent way. Using our notation, we examine simple co-evolutionary systems and discover new configurations by the simple process of "drawing on a whiteboard". Finally, we demonstrate that even complex, highly-interconnected systems with large numbers of populations can be understood with ease using the advanced features of our formalism
Evolutionary algorithms have been used in many ways to generate digital art. We study how evolutionary processes are used for evolutionary art and present a new approach to the transition of images. Our main idea is to define evolutionary processes for digital image transition, combining different variants of mutation and evolutionary mechanisms. We introduce box and strip mutation operators which are specifically designed for image transition. Our experimental results show that the process of an evolutionary algorithm in combination with these mutation operators can be used as a valuable way to produce unique generative art.
Chimera graphs define the topology of one of the first commercially available quantum computers. A variety of optimization problems have been mapped to this topology to evaluate the behavior of quantum enhanced optimization heuristics in relation to other optimizers, being able to efficiently solve problems classically to use them as benchmarks for quantum machines. In this paper we investigate for the first time the use of Evolutionary Algorithms (EAs) on Ising spin glass instances defined on the Chimera topology. Three genetic algorithms (GAs) and three estimation of distribution algorithms (EDAs) are evaluated over $1000$ hard instances of the Ising spin glass constructed from Sidon sets. We focus on determining whether the information about the topology of the graph can be used to improve the results of EAs and on identifying the characteristics of the Ising instances that influence the success rate of GAs and EDAs.
An important role carried out by cyber-security experts is the assessment of proposed computer systems, during their design stage. This task is fraught with difficulties and uncertainty, making the knowledge provided by human experts essential for successful assessment. Today, the increasing number of progressively complex systems has led to an urgent need to produce tools that support the expert-led process of system-security assessment. In this research, we use weighted averages (WAs) and ordered weighted averages (OWAs) with evolutionary algorithms (EAs) to create aggregation operators that model parts of the assessment process. We show how individual overall ratings for security components can be produced from ratings of their characteristics, and how these individual overall ratings can be aggregated to produce overall rankings of potential attacks on a system. As well as the identification of salient attacks and weak points in a prospective system, the proposed method also highlights which factors and security components contribute most to a component's difficulty and attack ranking respectively. A real world scenario is used in which experts were asked to rank a set of technical attacks, and to answer a series of questions about the security components that are the subject of the attacks. The work shows how finding good aggregation operators, and identifying important components and factors of a cyber-security problem can be automated. The resulting operators have the potential for use as decision aids for systems designers and cyber-security experts, increasing the amount of assessment that can be achieved with the limited resources available.
Many research works have successfully extended algorithms such as evolutionary algorithms, reinforcement agents and neural networks using "opposition-based learning" (OBL). Two types of the "opposites" have been defined in the literature, namely \textit{type-I} and \textit{type-II}. The former are linear in nature and applicable to the variable space, hence easy to calculate. On the other hand, type-II opposites capture the "oppositeness" in the output space. In fact, type-I opposites are considered a special case of type-II opposites where inputs and outputs have a linear relationship. However, in many real-world problems, inputs and outputs do in fact exhibit a nonlinear relationship. Therefore, type-II opposites are expected to be better in capturing the sense of "opposition" in terms of the input-output relation. In the absence of any knowledge about the problem at hand, there seems to be no intuitive way to calculate the type-II opposites. In this paper, we introduce an approach to learn type-II opposites from the given inputs and their outputs using the artificial neural networks (ANNs). We first perform \emph{opposition mining} on the sample data, and then use the mined data to learn the relationship between input $x$ and its opposite $\breve{x}$. We have validated our algorithm using various benchmark functions to compare it against an evolving fuzzy inference approach that has been recently introduced. The results show the better performance of a neural approach to learn the opposites. This will create new possibilities for integrating oppositional schemes within existing algorithms promising a potential increase in convergence speed and/or accuracy.
Perturb and Combine (P&C) group of methods generate multiple versions of the predictor by perturbing the training set or construction and then combining them into a single predictor (Breiman, 1996b). The motive is to improve the accuracy in unstable classification and regression methods. One of the most well known method in this group is Bagging. Arcing or Adaptive Resampling and Combining methods like AdaBoost are smarter variants of P&C methods. In this extended abstract, we lay the groundwork for a new family of methods under the P&C umbrella, known as Evolutionary Sampling (ES). We employ Evolutionary algorithms to suggest smarter sampling in both the feature space (sub-spaces) as well as training samples. We discuss multiple fitness functions to assess ensembles and empirically compare our performance against randomized sampling of training data and feature sub-spaces.
Over the years, several meta-heuristic algorithms were proposed and are now emerging as common methods for constrained optimization problems. Among them, genetic algorithms (GA's) shine as popular evolutionary algorithms (EA's) in engineering optimization. Most engineering design problems are difficult to resolve with conventional optimization algorithms because they are highly nonlinear and contain constraints. In order to handle these constraints, the most common technique is to apply penalty functions. The major drawback is that they require tuning of parameters, which can be very challenging. In this paper, we present a constraint-handling technique for GA's solely using the violation factor, called VCH (Violation Constraint-Handling) method. Several benchmark problems from the literature are examined. The VCH technique was able to provide a consistent performance and match results from other GA-based techniques.