Theoretical analyses of evolution strategies are indispensable for gaining a deep understanding of their inner workings. For constrained problems, rather simple problems are of interest in the current research. This work presents a theoretical analysis of a multi-recombinative evolution strategy with cumulative step size adaptation applied to a conically constrained linear optimization problem. The state of the strategy is modeled by random variables and a stochastic iterative mapping is introduced. For the analytical treatment, fluctuations are neglected and the mean value iterative system is considered. Nonlinear difference equations are derived based on one-generation progress rates. Based on that, expressions for the steady state of the mean value iterative system are derived. By comparison with real algorithm runs, it is shown that for the considered assumptions, the theoretical derivations are able to predict the dynamics and the steady state values of the real runs.

The behavior of the -Evolution Strategy (ES) with cumulative step size adaptation (CSA) on the ellipsoid model is investigated using dynamic systems analysis. At first a nonlinear system of difference equations is derived that describes the mean value evolution of the ES. This system is successively simplified to finally allow for deriving closed-form solutions of the steady state behavior in the asymptotic limit case of large search space dimensions. It is shown that the system exhibits linear convergence order. The steady state mutation strength is calculated, and it is shown that compared to standard settings in self-adaptive ESs, the CSA control rule allows for an approximately -fold larger mutation strength. This explains the superior performance of the CSA in non-noisy environments. The results are used to derive a formula for the expected running time. Conclusions regarding the choice of the cumulation parameter c and the damping constant D are drawn.

The convergence behaviors of so-called natural evolution strategies (NES) and of the information-geometric optimization (IGO) approach are considered. After a review of the NES/IGO ideas, which are based on information geometry, the implications of this philosophy w.r.t. optimization dynamics are investigated considering the optimization performance on the class of positive quadratic objective functions (the ellipsoid model). Exact differential equations describing the approach to the optimizer are derived and solved. It is rigorously shown that the original NES philosophy optimizing the expected value of the objective functions leads to very slow (i.e., sublinear) convergence toward the optimizer. This is the real reason why state of the art implementations of IGO algorithms optimize the expected value of transformed objective functions, for example, by utility functions based on ranking. It is shown that these utility functions are localized fitness functions that change during the IGO flow. The governing differential equations describing this flow are derived. In the case of convergence, the solutions to these equations exhibit an exponentially fast approach to the optimizer (i.e., linear convergence order). Furthermore, it is proven that the IGO philosophy leads to an adaptation of the covariance matrix that equals in the asymptotic limit—up to a scalar factor—the inverse of the Hessian of the objective function considered.

This paper studies the performance of multi-recombinative evolution strategies using isotropically distributed mutations with cumulative step length adaptation when applied to optimising cigar functions. Cigar functions are convex-quadratic objective functions that are characterised by the presence of only two distinct eigenvalues of their Hessian, the smaller one of which occurs with multiplicity one. A simplified model of the strategy's behaviour is developed. Using it, expressions that approximately describe the stationary state that is attained when the mutation strength is adapted are derived. The performance achieved by cumulative step length adaptation is compared with that obtained when using optimally adapted step lengths.

Evolutionary algorithms are frequently applied to dynamic optimization problems in which the objective varies with time. It is desirable to gain an improved understanding of the influence of different genetic operators and of the parameters of a strategy on its tracking performance. An approach that has proven useful in the past is to mathematically analyze the strategy's behavior in simple, idealized environments. The present paper investigates the performance of a multiparent evolution strategy that employs cumulative step length adaptation for an optimization task in which the target moves linearly with uniform speed. Scaling laws that quite accurately describe the behavior of the strategy and that greatly contribute to its understanding are derived. It is shown that in contrast to previously obtained results for a randomly moving target, cumulative step length adaptation fails to achieve optimal step lengths if the target moves in a linear fashion. Implications for the choice of population size parameters are discussed.

It is known that, in the absence of noise, no improvement in local performance can be gained from retaining candidate solutions other than the best one. Yet, it has been shown experimentally that, in the presence of noise, operating with a non-singular population of candidate solutions can have a marked and positive effect on the local performance of evolution strategies. So as to determine the reasons for the improved performance, we have studied the evolutionary dynamics of the (μ, λ)-ES in the presence of noise. Considering a simple, idealized environment, we have developed a moment-based approach that uses recent results involving concomitants of selected order statistics. This approach yields an intuitive explanation for the performance advantage of multi-parent strategies in the presence of noise. It is then shown that the idealized dynamic process considered does bear relevance to optimization problems in high-dimensional search spaces.

Cumulative step-size adaptation (CSA) based on path length control is regarded as a robust alternative to the standard mutative self-adaptation technique in evolution strategies (ES), guaranteeing an almost optimal control of the mutation operator. This paper shows that the underlying basic assumption in CSA — the perpendicularity of expected consecutive steps — does not necessarily guarantee optimal progress performance for (μ/μ I λ) intermediate recombinative ES

Self-adaptation is an essential feature of natural evolution. However, in the context of function optimization, self-adaptation features of evolutionary search algorithms have been explored mainly with evolution strategy (ES) and evolutionary programming (EP). In this paper, we demonstrate the self-adaptive feature of real-parameter genetic algorithms (GAs) using a simulated binary crossover (SBX) operator and without any mutation operator. The connection between the working of self-adaptive ESs and real-parameter GAs with the SBX operator is also discussed. Thereafter, the self-adaptive behavior of real-parameter GAs is demonstrated on a number of test problems commonly used in the ES literature. The remarkable similarity in the working principle of real-parameter GAs and self-adaptive ESs shown in this study suggests the need for emphasizing further studies on self-adaptive GAs.

The progress rate of the (1 +λ)-ES (Evolution Strategy) is analyzed on the parabolic ridge test function. A differentprogress behavior is observed for the (1, λ)-ES than for the sphere model test function. The characteristics of the progress rate picture for the plus strategy differs little from the one obtained for the sphere model, but this strategy has drastically worse progress rate values than those obtained for the comma strategy. The dynamics of the distance to the progress axis is also investigated. A theoretical formula is derived to estimate the change in this distance over generations. This formula is used to derive the expected value of the problem-specific distance to the ridge axis. The correctness of the formulae is supported by simulation results.

The progress behavior of evolution strategies (ES) using recombination is analyzed in this paper on the parabolic ridge. This test function represents landscapes far from the optimum. The ES algorithms with intermediate and dominant recombination are considered in the analysis. The derivations are presented for intermediate recombination. Thereafter, the formulae for dominant recombination are obtained using the so-called surrogate mutation model. In the analysis, the formulae are derived for the progress rate φ and for the stationary distance R (∞) to the ridge axis. As a result, it will be shown that the progress rate φ can increase if recombination is applied. Simulations are used to show the appropriateness of the formulae derived.

The effectiveness of intermediate recombination in evolution strategies is analyzed in light of the typical procedure of initializing trial solutions uniformly about the global optimum of benchmark functions. Analysis indicates that this procedure may predispose results in favor of intermediate recombination.

This paper analyzes the self-adaptation (SA) algorithm widely used to adapt strategy parameters of the evolution strategy (ES) in order to obtain maximal ES performance. The investigations are concentrated on the adaptation of one general mutation strength σ (called σSA) in (1, λ) ESs. The hypersphere serves as the fitness model. Starting from an introduction to the basic concept of self-adaptation, a framework for the analysis of σSA is developed on two levels: a microscopic level, concerning the description of the stochastic changes from one generation to the next, and a macroscopic level, describing the evolutionary dynamics of the σSA over time (generations). The σSA requires the fixing of a new strategy parameter, known as the learning parameter. The influence of this parameter on ES performance is investigated and rules for its tuning are presented and discussed. The results of the theoretical analysis are compared with ES experiments; it will be shown that applying Schwefel's τ-scaling rule guarantees the linear convergence order of the ES.

The multirecombinant (μ/μ, λ) evolution strategy (ES) is investigated for real-valued, N -dimensional parameter spaces. The analysis includes both intermediate recombination and dominant recombination, as well. These investigations are done for the spherical model first. The problem of the optimal population size depending on the parameter space dimension N is solved. A method extending the results obtained for the spherical model to nonspherical success domains is presented. The power of sexuality is discussed and it is shown that this power does not stem mainly from the “combination” of “good properties” of the mates (building block hypothesis) but rather from genetic repair diminishing the influence of harmful mutations. The dominant recombination is analyzed by introduction of surrogate mutations leading to the concept of species . Conclusions for evolutionary algorithms (EAs), including genetic algorithms (GAs), are drawn.

The multimembered evolution strategy (ES) acting on μ parents and λ offspring is analyzed for real-valued, N -dimensional parameter spaces ( N ≳ 30). N -dependent progress rate formulas are derived for (1, λ) and (μ, λ) strategies on spherical models. The analytical results obtained are compared with simulation experiments for the (hyper)sphere and the inclined (hyper)plane.

A method for the determination of the progress rate and the probability of success for the Evolution Strategy (ES) is presented. The new method is based on the asymptotical behavior of the χ-distribution and yields exact results in the case of infinite-dimensional parameter spaces. The technique is demonstrated for the (l, + λ) ES using a spherical model including noisy quality functions. The results are used to discuss the convergence behavior of the ES.