Michael Hellwig

The dynamics of cumulative step size adaptation on the ellipsoid model

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.

Evolution Under Strong Noise: A Self-Adaptive Evolution Strategy Can Reach the Lower Performance Bound - The pcCMSA-ES

According to a theorem by Astete-Morales, Cauwet, and Teytaud, “simple Evolution Strategies (ES)” that optimize quadratic functions disturbed by additive Gaussian noise of constant variance can only reach a simple regret log-log convergence slope \(\ge -1/2\) (lower bound). In this paper a population size controlled ES is presented that is able to perform better than the \(-1/2\) limit. It is shown experimentally that the pcCMSA-ES is able to reach a slope of \(-1\) being the theoretical lower bound of all comparison-based direct search algorithms.

Comparison of constraint-handling mechanisms for the 1,λ-es on a simple constrained problem

This paper investigates constraint-handling techniques used in nonelitist single-parent evolution strategies for the problem of maximizing a linear function with a single linear constraint. Two repair mechanisms are considered, and the analytical results are compared to those of earlier repair approaches in the same fitness environment. The first algorithm variant applies reflection to initially infeasible candidate solutions, and the second repair method uses truncation to generate feasible solutions from infeasible ones. The distributions describing the strategies’ one-generation behavior are calculated and used in a zeroth-order model for the steady state attained when operating with fixed step size. Considering cumulative step size adaptation, the qualitative differences in the behavior of the algorithm variants can be explained. The approach extends the theoretical knowledge of constraint-handling methods in the field of evolutionary computation and has implications for the design of constraint-handling techniques in connection with cumulative step size adaptation.

Mutation strength control by meta-ES on the sharp ridge

This paper investigates mutation strength control using Meta-ES on the sharp ridge. The asymptotical analysis presented allows for the prediction of the dynamics in ridge as well as in radial direction. Being based on this analysis the problem of the choice of population size λ and isolation parameter γ will be tackled. Remarkably, the qualitative convergence behavior is not determined by γ alone, but rather by the number of function evaluations λ γ devoted to the inner ES.

Analysis of the pcCMSA-ES on the noisy ellipsoid model

Regarding the noisy ellipsoid model with additive Gaussian noise, the population control covariance matrix self-adaptation Evolution Strategy (pcCMSA-ES) by Hellwig and Beyer was empirically observed to exhibit a convergence rate (CR) close to the theoretical lower bound of - 1 for all comparison-based direct search algorithms. The present paper provides the corresponding theoretical analysis of the pcCMSA-ES long-term behavior. To this end, the analysis from the context of isotropic mutations is transferred to the pcCMSA-ES that uses covariance matrix adaptation until significant noise influence is detected. The results allow for the computation of an upper bound on the number of generations between two consecutive test decisions of the pcCMSA-ES that ensures the observed performance. Further, the empirically observed convergence rate of CR ∼ −1 is theoretically derived.

Mutation strength control via meta evolution strategies on the ellipsoid model

The ability of a hierarchically organized evolution strategy (meta evolution strategy) with isolation periods of length one to optimally control its mutation strength is investigated on convex-quadratic functions (referred to as ellipsoid model). Applying the dynamical systems analysis approach a first step towards the analysis of the meta evolution strategy behavior is conducted. A non-linear system of difference equations is derived to describe the mean-value evolution of the respective hierarchically organized strategy. In the asymptotic limit case of large search space dimensions this system is suitable to derive closed-form solutions which describe the longterm behavior of the meta evolution strategy. The steady state mutation strength is bracketed within an interval depending on the mutation strength control parameter. Compared to standard settings in cumulative step-length adaptation evolution strategies the meta evolution strategy realizes almost similar normalized mutation strengths. The performance of the meta evolution strategy turns out to be very robust to the choice of its control parameters. The results allow for the derivation of the expected running time of the algorithm.

Controlling population size and mutation strength by Meta-ES under fitness noise

This paper investigates strategy parameter control by Meta-ES using the noisy sphere model. The fitness noise considered is normally distributed with constant noise variance. An asymptotical analysis concerning the mutation strength and the population size is presented. It allows for the prediction of the Meta-ES dynamics. An expression describing the asymptotical growth of the normalized mutation strength is calculated. Finally, the theoretical results are evaluated empirically.

On the Optimization of 2D Path Network Layouts in Engineering Designs via Evolutionary Computation Techniques

We describe an effective optimization strategy that is capable of discovering innovative cost-optimal designs of complete ascent assembly structures. Our approach relies on a continuous 2D model abstraction, an application-inspired multi-objective formulation of the optimal design task and an efficient coevolutionary solver. The obtained results provide empirical support that our novel strategy is able to deliver competitive results for the underlying general optimization challenge: the (obstacle-avoiding) Euclidean Steiner Tree Problem.

Optimization of Ascent Assembly Design Based on a Combinatorial Problem Representation

The paper addresses the integration of optimization in the automated design process of ascent assemblies. The goal is to automatically search for an optimal path connecting user defined inspection points while avoiding obstacles. As a first step towards full automation of the ascent assembly design, a discrete 2D model abstraction is considered. This establishes a combinatorial optimization problem, which is tackled by the use of two distinct strategies: a greedy heuristic and a genetic algorithm variant. Applying modeling approach and algorithms to multiple test cases, partly artificial and partly based on a manufactured crane, shows that the automated ascent assembly design tasks can successfully be enhanced with optimal path planning.

A Matrix Adaptation Evolution Strategy for Constrained Real-Parameter Optimization

By combination of successful constraint handling techniques known within the context of Differential Evolution with the recently suggested Matrix Adaptation Evolution Strategy (MA-ES), a new Evolution Strategy for constrained optimization is presented. The novel MA - ES variant is applied to the benchmark problems specified for the CEC 2018 competition on constrained single objective real-parameter optimization. The algorithm is able to find feasible solutions on more than 80 % of the benchmark problems with high accuracy.