We introduce linear surrogate functions for modeling inequality constraints to solve constrained blackbox optimization problems with the Augmented Lagrangian CMA-ES. Each surrogate is constructed from a binary classifier that predicts the sign of the constraint value. The classifier, and consequently the resulting algorithm, is invariant under sign preserving transformations of the constraint values and can handle binary, flat, and deceptive constraints. Somewhat surprisingly, we find that adopting a sign-based classification model of the constraints allows to solve classes of constrained problems which can not be solved with the original Augmented Lagrangian method using the true constraint value.

We present a surrogate-assisted multiobjective optimization algorithm. The aggregation of the objectives relies on the Uncrowded Hypervolume Improvement (UHVI) which is partly replaced by a linear-quadratic surrogate that is integrated into the CMA-ES algorithm. Surrogating the UHVI poses two challenges. First, the UHVI is a dynamic function, changing with the empirical Pareto set. Second, it is a composite function, defined differently for dominated and nondominated points. The presented algorithm is thought to be used with expensive functions of moderate dimension (up to about 50) with a quadratic surrogate which is updated based on its ranking ability. We report numerical experiments which include tests on the COCO benchmark. The algorithm shows in particular linear convergence on the double sphere function with a convergence rate that is 6--20 times faster than without surrogate assistance.

In this paper, we investigate the effect of a learning rate for the mean in Evolution Strategies with recombination. We study the effect of a half-line search after the mean shift direction is established, hence the learning rate value is conditioned to the direction. We prove convergence and study convergence rates in different dimensions and for different population sizes on the sphere function with the step-size proportional to the distance to the optimum.

Several test function suites are being used for numerical benchmarking of multiobjective optimization algorithms. While they have some desirable properties, such as well-understood Pareto sets and Pareto fronts of various shapes, most of the currently used functions possess characteristics that are arguably underrepresented in real-world problems such as separability, optima located exactly at the boundary constraints, and the existence of variables that solely control the distance between a solution and the Pareto front. Via the alternative construction of combining existing single-objective problems from the literature, we describe the bbob-biobj test suite with 55 bi-objective functions in continuous domain, and its extended version with 92 bi-objective functions (bbob-biobj-ext). Both test suites have been implemented in the COCO platform for black-box optimization benchmarking and various visualizations of the test functions are shown to reveal their properties. Besides providing details on the construction of these problems and presenting their (known) properties, this article also aims at giving the rationale behind our approach in terms of groups of functions with similar properties, objective space normalization, and problem instances. The latter allows us to easily compare the performance of deterministic and stochastic solvers, which is an often overlooked issue in benchmarking.

We present concepts and recipes for the anytime performance assessment when benchmarking optimization algorithms in a blackbox scenario. We consider runtime—oftentimes measured in the number of blackbox evaluations needed to reach a target quality—to be a universally measurable cost for solving a problem. Starting from the graph that depicts the solution quality versus runtime, we argue that runtime is the only performance measure with a generic, meaningful, and quantitative interpretation. Hence, our assessment is solely based on runtime measurements. We discuss proper choices for solution quality indicators in single- and multi-objective optimization, as well as in the presence of noise and constraints. We also discuss the choice of the target values, budget-based targets, and the aggregation of runtimes by using simulated restarts, averages, and empirical cumulative distributions which generalize convergence graphs of single runs. The presented performance assessment is to a large extent implemented in the comparing continuous optimizers (COCO) platform freely available athttps://github.com/numbbo/coco.

In this paper, we investigate a non-elitist Evolution Strategy designed to handle black-box constraints by an adaptive Augmented Lagrangian penalty approach, AL-(μ/μw, λ)-CMA-ES, on problems with up to 28 constraints. Based on stability and performance observations, we propose an improved default parameter setting. We exhibit failure cases of the Augmented Lagrangian technique and show how surrogate modeling of the constraints can overcome some difficulties. Several variants of AL-CMA-ES are compared on a set of nonlinear constrained problems from the literature. Simple adaptive penalty techniques serve as a baseline for comparison.

The hypervolume indicator is widely used by multi-objective optimization algorithms and for assessing their performance. We investigate a set of p vectors in the biobjective space that maximizes the hypervolume indicator with respect to some reference point, referred to as p-optimal distribution. We prove explicit lower and upper bounds on the gap between the hypervolumes of the p-optimal distribution and the p-optimal distribution (the Pareto front) as a function of p, of the reference point, and of some Lipschitz constants. On a wide class of functions, this optimality gap can not be smaller than p(1/p), thereby establishing a bound on the optimal convergence speed of any algorithm. For functions with either bilipschitz or convex Pareto fronts, we also establish an upper bound and the gap is hence Θ(1/p). The presented bounds are not only asymptotic. In particular, functions with a linear Pareto front have the normalized exact gap of 1/(p + 1) for any reference point dominating the nadir point.

We introduce an acceleration for covariance matrix adaptation evolution strategies (CMA-ES) by means of adaptive diagonal decoding (dd-CMA). This diagonal acceleration endows the default CMA-ES with the advantages of separable CMA-ES without inheriting its drawbacks. Technically, we introduce a diagonal matrix [Formula: see text] that expresses coordinate-wise variances of the sampling distribution in DCD form. The diagonal matrix can learn a rescaling of the problem in the coordinates within a linear number of function evaluations. Diagonal decoding can also exploit separability of the problem, but, crucially, does not compromise the performance on nonseparable problems. The latter is accomplished by modulating the learning rate for the diagonal matrix based on the condition number of the underlying correlation matrix. dd-CMA-ES not only combines the advantages of default and separable CMA-ES, but may achieve overadditive speedup: it improves the performance, and even the scaling, of the better of default and separable CMA-ES on classes of nonseparable test functions that reflect, arguably, a landscape feature commonly observed in practice. The article makes two further secondary contributions: we introduce two different approaches to guarantee positive definiteness of the covariance matrix with active CMA, which is valuable in particular with large population size; we revise the default parameter setting in CMA-ES, proposing accelerated settings in particular for large dimension. All our contributions can be viewed as independent improvements of CMA-ES, yet they are also complementary and can be seamlessly combined. In numerical experiments with dd-CMA-ES up to dimension 5120, we observe remarkable improvements over the original covariance matrix adaptation on functions with coordinate-wise ill-conditioning. The improvement is observed also for large population sizes up to about dimension squared.

Quality gain is the expected relative improvement of the function value in a single step of a search algorithm. Quality gain analysis reveals the dependencies of the quality gain on the parameters of a search algorithm, based on which one can derive the optimal values for the parameters. In this paper, we investigate evolution strategies with weighted recombination on general convex quadratic functions. We derive a bound for the quality gain and two limit expressions of the quality gain. From the limit expressions, we derive the optimal recombination weights and the optimal step-size, and find that the optimal recombination weights are independent of the Hessian of the objective function. Moreover, the dependencies of the optimal parameters on the dimension and the population size are revealed. Differently from previous works where the population size is implicitly assumed to be smaller than the dimension, our results cover the population size proportional to or greater than the dimension. Numerical simulation shows that the asymptotically optimal step-size well approximates the empirically optimal step-size for a finite dimensional convex quadratic function.

We explore the arguably simplest way to build an effective surrogate fitness model in continuous search spaces. The model complexity is linear or diagonal-quadratic or full quadratic, depending on the number of available data. The model parameters are computed from the Moore-Penrose pseudoinverse. The model is used as a surrogate fitness for CMA-ES if the rank correlation between true fitness and surrogate value of recently sampled data points is high. Otherwise, further samples from the current population are successively added as data to the model. We empirically compare the IPOP scheme of the new model assisted lq-CMA-ES with a variety of previously proposed methods and with a simple portfolio algorithm using SLSQP and CMA-ES. We conclude that a global quadratic model and a simple portfolio algorithm are viable options to enhance CMA-ES. The model building code is available as part of the pycma Python module on Github and PyPI.

We present a framework to build a multiobjective algorithm from single-objective ones. This framework addresses the p × n-dimensional problem of finding p solutions in an n-dimensional search space, maximizing an indicator by dynamic subspace optimization. Each single-objective algorithm optimizes the indicator function given p - 1 fixed solutions. Crucially, dominated solutions minimize their distance to the empirical Pareto front defined by these p - 1 solutions. We instantiate the framework with CMA-ES as single-objective optimizer. The new algorithm, COMO-CMA-ES, is empirically shown to converge linearly on bi-objective convex-quadratic problems and is compared to MO-CMA-ES, NSGA-II and SMS-EMOA.

We introduce two suites of mixed-integer benchmark problems to be used for analyzing and comparing black-box optimization algorithms. They contain problems of diverse difficulties that are scalable in the number of decision variables. The bbob-mixint suite is designed by partially discretizing the established BBOB (Black-Box Optimization Benchmarking) problems. The bi-objective problems from the bbob-biobj-mixint suite are, on the other hand, constructed by using the bbob-mixint functions as their separate objectives. We explain the rationale behind our design decisions and show how to use the suites within the COCO (Comparing Continuous Optimizers) platform. Analyzing two chosen functions in more detail, we also provide some unexpected findings about their properties.

The CMA-ES is one of the most powerful stochastic numerical optimizers to address difficult black-box problems. Its intrinsic time and space complexity is quadratic—limiting its applicability with increasing problem dimensionality. To circumvent this limitation, different large-scale variants of CMA-ES with subquadratic complexity have been proposed over the past ten years. To-date however, these variants have been tested and compared only in rather restrictive settings, due to the lack of a comprehensive large-scale testbed to assess their performance. In this context, we introduce a new large-scale testbed with dimension up to 640, implemented within the COCO benchmarking platform. We use this testbed to assess the performance of several promising variants of CMA-ES and the standard limited-memory L-BFGS. In all tested dimensions, the best CMA-ES variant solves more problems than L-BFGS for larger budgets while L-BFGS outperforms the best CMA-ES variant for smaller budgets. However, over all functions, the cumulative runtime distributions between L-BFGS and the best CMA-ES variants are close (less than a factor of 4 in high dimension). Our results illustrate different scaling behaviors of the methods, expose a few defects of the algorithms and reveal that for dimension larger than 80, LM-CMA solves more problems than VkD-CMA while in the cumulative runtime distribution over all functions the VkD-CMA dominates LM-CMA for budgets up to $$10^4$$ times dimension and for all budgets up to dimension 80.

We investigate evolution strategies with weighted recombination on general convex quadratic functions. We derive the asymptotic quality gain in the limit of the dimension to infinity, and derive the optimal recombination weights and the optimal step-size. This work is an extension of previous works where the asymptotic quality gain of evolution strategies with weighted recombination was derived on the infinite dimensional sphere function. Moreover, for a finite dimensional search space, we derive rigorous bounds for the quality gain on a general quadratic function. They reveal the dependency of the quality gain both in the eigenvalue distribution of the Hessian matrix and on the recombination weights. Taking the search space dimension to infinity, it turns out that the optimal recombination weights are independent of the Hessian matrix, i.e., the recombination weights optimal for the sphere function are optimal for convex quadratic functions.

We analyze linear convergence of an evolution strategy for constrained optimization with an augmented Lagrangian constraint handling approach. We study the case of multiple active linear constraints and use a Markov chain approach---used to analyze randomized optimization algorithms in the unconstrained case---to establish linear convergence under sufficient conditions. More specifically, we exhibit a class of functions on which a homogeneous Markov chain (defined from the state variables of the algorithm) exists and whose stability implies linear convergence. This class of functions is defined such that the augmented Lagrangian, centered in its value at the optimum and the associated Lagrange multipliers, is positive homogeneous of degree $2$, and includes convex quadratic functions. Simulations of the Markov chain are conducted on linearly constrained sphere and ellipsoid functions to validate numerically the stability of the constructed Markov chain.

We present a canonical way to turn any smooth parametric family of probability distributions on an arbitrary search space $X$ into a continuous-time black-box optimization method on $X$, the information-geometric optimization (IGO) method. Invariance as a major design principle keeps the number of arbitrary choices to a minimum. The resulting IGO flow is the flow of an ordinary differential equation conducting the natural gradient ascent of an adaptive, time-dependent transformation of the objective function. It makes no particular assumptions on the objective function to be optimized. The IGO method produces explicit IGO algorithms through time discretization. It naturally recovers versions of known algorithms and offers a systematic way to derive new ones. In continuous search spaces, IGO algorithms take a form related to natural evolution strategies (NES). The cross-entropy method is recovered in a particular case with a large time step, and can be extended into a smoothed, parametrization-independent maximum likelihood update (IGO-ML). When applied to the family of Gaussian distributions on $\R^d$, the IGO framework recovers a version of the well-known CMA-ES algorithm and of xNES. For the family of Bernoulli distributions on $\{0,1\}^d$, we recover the seminal PBIL algorithm and cGA. For the distributions of restricted Boltzmann machines, we naturally obtain a novel algorithm for discrete optimization on $\{0,1\}^d$. All these algorithms are natural instances of, and unified under, the single information-geometric optimization framework. The IGO method achieves, thanks to its intrinsic formulation, maximal invariance properties: invariance under reparametrization of the search space $X$, under a change of parameters of the probability distribution, and under increasing transformation of the function to be optimized. The latter is achieved through an adaptive, quantile-based formulation of the objective. Theoretical considerations strongly suggest that IGO algorithms are essentially characterized by a minimal change of the distribution over time. Therefore they have minimal loss in diversity through the course of optimization, provided the initial diversity is high. First experiments using restricted Boltzmann machines confirm this insight. As a simple consequence, IGO seems to provide, from information theory, an elegant way to simultaneously explore several valleys of a fitness landscape in a single run.

We propose a novel variant of the covariance matrix adaptation evolution strategy (CMA-ES) using a covariance matrix parameterized with a smaller number of parameters. The motivation of a restricted covariance matrix is twofold. First, it requires less internal time and space complexity that is desired when optimizing a function on a high dimensional search space. Second, it requires less function evaluations to adapt the covariance matrix if the restricted covariance matrix is rich enough to express the variable dependencies of the problem. In this paper we derive a computationally efficient way to update the restricted covariance matrix where the model richness of the covariance matrix is controlled by an integer and the internal complexity per function evaluation is linear in this integer times the dimension, compared to quadratic in the dimension in the CMA-ES. We prove that the proposed algorithm is equivalent to the sep-CMA-ES if the covariance matrix is restricted to the diagonal matrix, it is equivalent to the original CMA-ES if the matrix is not restricted. Experimental results reveal the class of efficiently solvable functions depending on the model richness of the covariance matrix and the speedup over the CMA-ES.

We address the question of linear convergence of evolution strategies on constrained optimization problems. In particular, we analyze a (1+1)-ES with an augmented Lagrangian constraint handling approach on functions defined on a continuous domain, subject to a single linear inequality constraint. We identify a class of functions for which it is possible to construct a homogeneous Markov chain whose stability implies linear convergence. This class includes all functions such that the augmented Lagrangian of the problem, centered with respect to its value at the optimum and the corresponding Lagrange multiplier, is positive homogeneous of degree 2 (thus including convex quadratic functions as a particular case). The stability of the constructed Markov chain is empirically investigated on the sphere function and on a moderately ill-conditioned ellipsoid function.

We propose a general methodology to construct large-scale testbeds for the benchmarking of continuous optimization algorithms. Our approach applies an orthogonal transformation on raw functions that involve only a linear number of operations. The orthogonal transformation is sampled from a parametrized family of transformations that are the product of a permutation matrix times a block-diagonal matrix times a permutation matrix. We investigate the impact of the different parameters of the transformation on the difficulty of the problems using the separable CMA-ES. We illustrate the use of the above defined transformation in the BBOB-2009 testbed as replacement for the expensive orthogonal (rotation) matrices. We also show the practicability of the approach by studying the computational cost and its applicability in a large scale setting.

Algorithm benchmarking plays a vital role in designing new optimization algorithms and in recommending efficient and robust algorithms for practical purposes. So far, two main approaches have been used to compare algorithms in the evolutionary multiobjective optimization (EMO) field: (i) displaying empirical attainment functions and (ii) reporting statistics on quality indicator values. Most of the time, EMO benchmarking studies compare algorithms for fixed and often arbitrary budgets of function evaluations although the algorithms are any-time optimizers. Instead, we propose to transfer and adapt standard benchmarking techniques from the single-objective optimization and classical derivative-free optimization community to the field of EMO. Reporting \emph{target-based runlengths} allows to compare algorithms with varying numbers of function evaluations quantitatively. Displaying data profiles can aggregate performance information over different test functions, problem difficulties, and quality indicators. We apply this approach to compare three common algorithms on a new test function suite derived from the well-known single-objective BBOB functions. The focus thereby lies less on gaining insights into the algorithms but more on showcasing the concepts and on what can be gained over current benchmarking approaches.

This paper analyzes a (1, λ)-Evolution Strategy, a randomized comparison-based adaptive search algorithm optimizing a linear function with a linear constraint. The algorithm uses resampling to handle the constraint. Two cases are investigated: first, the case where the step-size is constant, and second, the case where the step-size is adapted using cumulative step-size adaptation. We exhibit for each case a Markov chain describing the behavior of the algorithm. Stability of the chain implies, by applying a law of large numbers, either convergence or divergence of the algorithm. Divergence is the desired behavior. In the constant step-size case, we show stability of the Markov chain and prove the divergence of the algorithm. In the cumulative step-size adaptation case, we prove stability of the Markov chain in the simplified case where the cumulation parameter equals 1, and discuss steps to obtain similar results for the full (default) algorithm where the cumulation parameter is smaller than 1. The stability of the Markov chain allows us to deduce geometric divergence or convergence, depending on the dimension, constraint angle, population size, and damping parameter, at a rate that we estimate. Our results complement previous studies where stability was assumed.

This paper analyses a (1, λ)-Evolution Strategy, a randomised comparison-based adaptive search algorithm, on a simple constraint optimization problem. The algorithm uses resampling to handle the constraint and optimizes a linear function with a linear constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using path length control. We exhibit for each case a Markov chain whose stability analysis would allow us to deduce the divergence of the algorithm depending on its internal parameters. We show divergence at a constant rate when the step-size is constant. We sketch that with step-size adaptation geometric divergence takes place. Our results complement previous studies where stability was assumed.

We propose a novel natural gradient based stochastic search algorithm, VD-CMA, for the optimization of high dimensional numerical functions. The algorithm is comparison-based and hence invariant to monotonic transformations of the objective function. It adapts a multivariate normal distribution with a restricted covariance matrix with twice the dimension as degrees of freedom, representing an arbitrarily oriented long axis and additional axis-parallel scaling. We derive the different components of the algorithm and show linear internal time and space complexity. We find empirically that the algorithm adapts its covariance matrix to the inverse Hessian on convex-quadratic functions with an Hessian with one short axis and different scaling on the diagonal. We then evaluate VD-CMA on test functions and compare it to different methods. On functions covered by the internal model of VD-CMA and on the Rosenbrock function, VD-CMA outperforms CMA-ES (having quadratic internal time and space complexity) not only in internal complexity but also in number of function calls with increasing dimension.

Success rule based step-size adaptation, namely the one-fifth success rule, has shown to be effective for single parent evolution strategies (ES), e.g. the (1+1)-ES. The success rule remains feasible in non-elitist single parent strategies, where the target success rate must be roughly inversely proportional to the population size. This success rule is, however, not easily applicable to multi-parent strategies. In this paper, we introduce the median success rule for step-size adaptation, applicable to non-elitist multi-recombinant evolution strategies. The median success rule compares the median fitness of the population to a fitness from the previous iteration. The comparison fitness is chosen to achieve a target success rate of 1/2, thereby a deviation from the target can be measured reliably in comparatively few iteration steps. As a prerequisite for feasibility of the median success rule, we studied the way the fitness comparison quantile depends on the search space dimension, the population size, the parent number, the recombination weights and the objective function. The findings are encouraging: the choice of the comparison quantile appears to be relatively uncritical and experiments on a variety of functions, also in combination with CMA, reveal reasonable behavior.

This paper introduces a novel constraint handling approach for covariance matrix adaptation evolution strategies (CMA-ES). The key idea is to approximate the directions of the local normal vectors of the constraint boundaries by accumulating steps that violate the respective constraints, and to then reduce variances of the mutation distribution in those directions. The resulting strategy is able to approach the boundary of the feasible region without being impeded in its ability to search in directions tangential to the boundaries. The approach is implemented in the (1+1)-CMA-ES and evaluated numerically on several test problems. The results compare very favourably with data for other constraint handling approaches applied to unimodal test problems that can be found in the literature.

Benchmarking of optimization algorithms is necessary to quantitatively assess the performance of optimizers and to understand their strengths and weaknesses. The Black Box Optimization Benchmarking (BBOB) workshops that took place in 2009, 2010, and 2012 during the Genetic and Evolutionary Computation Conference (GECCO) were set up to benchmark both stochastic and deterministic continuous optimization algorithms. For this purpose, a thorough experimental setting, a set of test functions, and a visualization tool were designed and provided. They are based on the idea that (i) test functions should be representative of typical known difficulties, scalable with dimension, and not too easy to solve, yet comprehensible; and (ii) performance measures should be quantitative. A tool for acquiring and postprocessing data was provided.This special issue on Black Box Optimization Benchmarking contains papers that are extensions or based on results obtained during the BBOB GECCO 2009 and 2010 workshops. All articles were selected after the standard rigorous review process, from which seven papers in total were selected; five are published in this special issue and two papers will appear–because of space reasons–in a regular issue.We would like to thank all of the authors for contributing to the special issue as well as the reviewers for their reviews. We are indebted to Hans-Georg Beyer, Editor-in-Chief of Evolutionary Computation, for his patience and support. The works presented in this special issue rely heavily on the Comparing Continuous Optimizer (COCO) tool continuously developed since 2008 by the BBOB team among which we would like to thank in particular for their work and enthusiasm Raymond Ros, Steffen Finck, Petr Po.šì.k, Mike Preuss, Olaf Mersmann, and Verena Heidrich-Meisner.

This paper introduces mirrored sampling into evolution strategies (ESs) with weighted multi-recombination. Two further heuristics are introduced: pairwise selection selects at most one of two mirrored vectors in order to avoid a bias due to recombination. Selective mirroring only mirrors the worst solutions of the population. Convergence rates on the sphere function are derived that also yield upper bounds for the convergence rate on any spherical function. The optimal fraction of offspring to be mirrored is regardless of pairwise selection one without selective mirroring and about 19% with selective mirroring, where the convergence rate reaches a value of 0.390. This is an improvement of 56% compared to the best known convergence rate of 0.25 with positive recombination weights.

We propose a novel variant of the (1+1)-CMA-ES that updates the distribution of mutation vectors based on both successful and unsuccessful trial steps. The computational costs of the adaptation procedure are quadratic in the dimensionality of the problem, and the algorithm retains all invariance properties. Its performance on a set of standard test functions is compared with that of the original strategy that updates the distribution of mutation vectors in response to successful steps only. The new variant is not observed to be more than marginally slower on any function, and it is up to two times faster on some of the test problems.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We present a novel method for handling uncertainty in evolutionary optimization. The method entails quantification and treatment of uncertainty and relies on the rank based selection operator of evolutionary algorithms. The proposed uncertainty handling is implemented in the context of the covariance matrix adaptation evolution strategy (CMA-ES) and verified on test functions. The present method is independent of the uncertainty distribution, prevents premature convergence of the evolution strategy and is well suited for online optimization as it requires only a small number of additional function evaluations. The algorithm is applied in an experimental setup to the online optimization of feedback controllers of thermoacoustic instabilities of gas turbine combustors. In order to mitigate these instabilities, gain-delay or model-basedHinfincontrollers sense the pressure and command secondary fuel injectors. The parameters of these controllers are usually specified via a trial and error procedure. We demonstrate that their online optimization with the proposed methodology enhances, in an automated fashion, the online performance of the controllers, even under highly unsteady operating conditions, and it also compensates for uncertainties in the model-building and design process.

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.

This paper investigates the limits of the predictions based on the classical progress rate theory for Evolution Strategies. We explain on the sphere function why positive progress rates give convergence in mean, negative progress rates divergence in mean and show that almost sure convergence can take place despite divergence in mean. Hence step-sizes associated to negative progress can actually lead to almost sure convergence. Based on these results we provide an alternative progress rate definition related to almost sure convergence. We present Monte Carlo simulations to investigate the discrepancy between both progress rates and therefore both types of convergence. This discrepancy vanishes when dimension increases. The observation is supported by an asymptotic estimation of the new progress rate definition.

First, the covariance matrix adaptation (CMA) with rank-one update is introduced into the (1+1)-evolution strategy. An improved implementation of the 1/5-th success rule is proposed for step size adaptation, which replaces cumulative path length control. Second, an incremental Cholesky update for the covariance matrix is developed replacing the computational demanding and numerically involved decomposition of the covariance matrix. The Cholesky update can replace the decomposition only for the update without evolution path and reduces the computational effort from O(n3) to O(n2). The resulting (1+1)-Cholesky-CMA-ES is an elegant algorithm and the perhaps simplest evolution strategy with covariance matrix and step size adaptation. Simulations compare the introduced algorithms to previously published CMA versions.

This paper investigates sigma-self-adaptation for real valued evolutionary algorithms on linear fitness functions. We identify the step-size logarithm log sigma as a key quantity to understand strategy behavior. Knowing the bias of mutation, recombination, and selection on log sigma is sufficient to explain sigma-dynamics and strategy behavior in many cases, even from previously reported results on non-linear and/or noisy fitness functions. On a linear fitness function, if intermediate multi-recombination is applied on the object parameters, the i-th best and the i-th worst individual have the same sigma-distribution. Consequently, the correlation between fitness and step-size sigma is zero. Assuming additionally that sigma-changes due to mutation and recombination are unbiased, then sigma-self-adaptation enlarges sigma if and only if mu < lambda/2, given (mu, lambda)-truncation selection. Experiments show the relevance of the given assumptions.

In this paper we introduce a restart-CMA-evolution strategy, where the population size is increased for each restart (IPOP). By increasing the population size the search characteristic becomes more global after each restart. The IPOP-CMA-ES is evaluated on the test suit of 25 functions designed for the special session on real-parameter optimization of CEC 2005. Its performance is compared to a local restart strategy with constant small population size. On unimodal functions the performance is similar. On multi-modal functions the local restart strategy significantly outperforms IPOP in 4 test cases whereas IPOP performs significantly better in 29 out of 60 tested cases.

One natural question when testing performance of global optimization algorithm is: how performances compare to a restart local search algorithm. One purpose of this paper is to provide results for such comparisons. To this end, the performances of a restart (advanced) local-search strategy, the CMA-ES with small initial step-size, are investigated on the 25 functions of the CEC 2005 real-parameter optimization test suit. The second aim is to clarify the theoretical background of the performance criterion proposed to quantitatively compare the search algorithms. The theoretical analysis allows us to generalize the criterion proposed and to define a new criterion that can be applied more appropriate in a different context.

This paper presents a novel evolutionary optimization strategy based on the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). This new approach is intended to reduce the number of generations required for convergence to the optimum. Reducing the number of generations, i.e., the time complexity of the algorithm, is important if a large population size is desired: (1) to reduce the effect of noise; (2) to improve global search properties; and (3) to implement the algorithm on (highly) parallel machines. Our method results in a highly parallel algorithm which scales favorably with large numbers of processors. This is accomplished by efficiently incorporating the available information from a large population, thus significantly reducing the number of generations needed to adapt the covariance matrix. The original version of the CMA-ES was designed to reliably adapt the covariance matrix in small populations but it cannot exploit large populations efficiently. Our modifications scale up the efficiency to population sizes of up to 10n, where n is the problem dimension. This method has been applied to a large number of test problems, demonstrating that in many cases the CMA-ES can be advanced from quadratic to linear time complexity.

This paper puts forward two useful methods for self-adaptation of the mutation distribution - the concepts of derandomization and cumulation. Principle shortcomings of the concept of mutative strategy parameter control and two levels of derandomization are reviewed. Basic demands on the self-adaptation of arbitrary (normal) mutation distributions are developed. Applying arbitrary, normal mutation distributions is equivalent to applying a general, linear problem encoding. The underlying objective of mutative strategy parameter control is roughly to favor previously selected mutation steps in the future. If this objective is pursued rigorously, a completely derandomized self-adaptation scheme results, which adapts arbitrary normal mutation distributions. This scheme, called covariance matrix adaptation (CMA), meets the previously stated demands. It can still be considerably improved by cumulation - utilizing an evolution path rather than single search steps. Simulations on various test functions reveal local and global search properties of the evolution strategy with and without covariance matrix adaptation. Their performances are comparable only on perfectly scaled functions. On badly scaled, non-separable functions usually a speed up factor of several orders of magnitude is observed. On moderately mis-scaled functions a speed up factor of three to ten can be expected.

Comparable to other optimization techniques, the performance of evolution strategies (ESs) depends on a suitable choice of internal strategy control parameters. Apart from a fixed setting, ESs facilitate an adjustment of such parameters within a self-adaptation process. For step-size control in particular, various adaptation concepts have been evolved early in the development of ESs. These algorithms mostly work very efficiently as long as the scaling of the parameters to be optimized is known. If the scaling is not known, the strategy has to adapt individual step-sizes for all the parameters. In general, the number of necessary step-sizes (variances) equals the dimension of the problem. In this case, step-size adaptation proves to be difficult, and the algorithms known are not satisfactory. The algorithm presented in this paper is based on the well-known concept of mutative step-size control. Our investigations indicate that the adaptation by this concept declines due to an interaction of the random elements involved. We show that this weak point of mutative step-size control can be avoided by relatively small changes in the algorithm. The modifications may be summarized by the word “derandomization.” The derandomized scheme of mutative step-size control facilitates a reliable self-adaptation of individual step-sizes.