Abstract—Evolutionary algorithms often have to solve optimization problems in the presence of a wide range of uncertainties. Generally, uncertainties in evolutionary computation can be divided into the following four categories. First, the fitness function is noisy. Second, the design variables and/or the environmental parameters may change after optimization, and the quality of the obtained optimal solution should be robust against environmental changes or deviations from the optimal point. Third, the fitness function is approximated, which means that the fitness function suffers from approximation errors. Fourth, the optimum of the problem to be solved changes over time and, thus, the optimizer should be able to track the optimum continuously. In all these cases, additional measures must be taken so that evolutionary algorithms are still able to work satisfactorily. This paper attempts to provide a comprehensive overview of the related work within a unified framework, which has been scattered in a variety of research areas. Existing approaches to addressing different uncertainties are presented and discussed, and the relationship between the different categories of uncertainties are investigated. Finally, topics for future research are suggested. Index Terms—Approximation models, dynamic environments, noise, robustness, uncertainty. I.

Abstract — 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 set-up to the online optimization of feedback controllers of thermoacoustic instabilities of gas turbine combustors. In order to mitigate these instabilities, gain-delay or model-based H ∞ controllers 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. I.

This paper presents a simple multimembered evolution strategy (SMES) to solve global nonlinear optimization problems. The approach does not require the use of a penalty function and it does not require any extra parameters (besides those used with an evolution strategy). Instead, it uses a simple diversity mechanism based on allowing infeasible solutions to remain in the population This technique helps the algorithm to find the global optimum despite reaching reasonably fast the feasible region of the search space. Some simple selection criteria are used to guide the process to the feasible region of the search space. Also, the initial step size of the evolution strategy is reduced in order to perform a finer search and a combined (discrete/intermediate) recombination technique improves its exploitation capabilities. The approach was tested with a well-known benchmark. The results obtained are very competitive, when comparing the proposed approach against other state-of-the art techniques and its computational cost (measured by the number of fitness function evaluations) is lower than the required cost of the other techniques compared. 1

Abstract—For many real-world optimization problems, the robustness of a solution is of great importance in addition to the solution’s quality. By robustness, we mean that small deviations from the original design, e.g., due to manufacturing tolerances, should be tolerated without a severe loss of quality. One way to achieve that goal is to evaluate each solution under a number of different scenarios and use the average solution quality as fitness. However, this approach is often impractical, because the cost for evaluating each individual several times is unacceptable. In this paper, we present a new and efficient approach to estimating a solution’s expected quality and variance. We propose to construct local approximate models of the fitness function and then use these approximate models to estimate expected fitness and variance. Based on a variety of test functions, we demonstrate empirically that our approach significantly outperforms the implicit averaging approach, as well as the explicit averaging approaches using existing estimation techniques reported in the literature. Index Terms—Evolutionary optimization, fitness approximation, robustness, uncertainty. I.

This paper proposes and analyzes a class of test functions for evolutionary robust optimization, the so-called "Functions with Noise-Induced Multi-Modality" (FNIMs). After a motivational introduction gleaned from a real-world optimization problem, the robust optimizer properties of this test class are investigated w.r.t. different robustness measures. The steady state behavior of Evolution Strategies (ES) on FNIMs will be investigated empirically. Being based on the empirical results, a subclass of FNIMs is identified which is amenable to an asymptotical performance analysis. The results of this analysis will be used to derive recommendations for the choice of strategy-specific parameters such as population size and truncation ratio.

Abstract. The method of differential-geometry is applied for deriving steady state conditions for the (µ/µI,λ)-ES on the general quadratic test function disturbed by fitness noise of constant strength. A new approach for estimating the expected final fitness deviation observed under such conditions is presented. The theoretical results obtained are compared with real ES runs showing a surprisingly excellent agreement. 1

Abstract. We consider evolutionary model selection for support vector machines. Hold-out set-based objective functions are natural model selection criteria, and we introduce a symmetrization of the standard cross-validation approach. We propose the covariance matrix adaptation evolution strategy (CMA-ES) with uncertainty handling for optimizing the new randomized objective function. Our results show that this search strategy avoids premature convergence and results in improved classification accuracy compared to strategies without uncertainty handling. 1

Abstract. We investigate (1,λ) ESs using isotropic mutations for optimization inR n by means of a theoretical runtime analysis. In particular, a constant offspring-population size λ will be of interest. We start off by considering an adaptation-less (1,2) ES minimizing a linear function. Subsequently, a piecewise linear function with a jump/cliff is considered, where a (1+λ) ES gets trapped, i. e., (at least) an exponential (in n) number of steps are necessary to escape the local-optimum region. The (1,2) ES, however, manages to overcome the cliff in an almost unnoticeable number of steps. Finally, we outline (because of the page limit) how the reasoning and the calculations can be extended to the scenario where a (1,λ) ES using Gaussian mutations minimizes Cliff, a bimodal, spherically symmetric function already considered in the literature, which is merely Sphere with a jump in the function value at a certain distance from the minimum. For λ a constant large enough, the (1,λ) ES manages to conquer the global-optimum region – in contrast to (1+λ) ESs which get trapped. 1

Many variants of evolutionary algorithms have been designed and applied. The experimental knowledge is immense. The rigorous analysis of evolutionary algorithms is difficult, but such a theory can help to understand, design, and teach evolutionary algorithms. In this survey, first the history of attempts to analyse evolutionary algorithms is described and then new methods for continuous as well as discrete search spaces are presented and discussed.

In this paper, we argue that the original self-adaptation mechanism of the Evolution Strategies is useful by itself to handle constraints in global optimization. We show how using just three simple comparison criteria the simple Evolution Strategy can be led to the feasible region of the search space and find the global optimum solution (or a very good approximation of it). Different Evolution Strategies including 65 50 with or without correlated mutation were implemented. Such approaches have been tested using the well-known test suit of Michalewicz and Schnoenauer and four engineering problems. The results are discussed and some conclusions are drawn. 1