How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms

Abstract

The (1+1) evolution strategy (ES), a simple, mutation-based evolutionary algorithm for continuous optimization problems, is analyzed. In particular, we consider the most common type of mutations, namely Gaussian mutations, and the 1 5 -rule for mutation adaptation, and we are interested in how the runtime/number of function evaluations to obtain a predefined reduction of the approximation error depends on the dimension of the search space.The most discussed function in the area of ES is the so-called SPHERE-function given by SPHERE: R n → R with x → x Ix (where I ∈ R n×n is the identity matrix), which also has already been the subject of a runtime analysis. This analysis is extended to arbitrary positive definite quadratic forms that induce ellipsoidal fitness landscapes which are "close to being spherically symmetric", showing that the order of the runtime does not change compared to SPHERE. Furthermore, certain positive definite quadratic forms f : R n → R with x → x Qx, where Q ∈ R n×n , inducing ellipsoidal fitness landscapes that are "far away from being spherically symmetric" are exemplarily investigated, namely f (x) = · x 2 1 + · · · + x 2 n/2 + x 2 n/2+1 + · · · + x 2 n with = poly(n) such that 1/ → 0 as n → ∞. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime to halve the approximation error is ( · n) compared to (n) for SPHERE.

Cooperative differential evolution with fast variable interdependence learning and cross-cluster mutation

“…The widely used optimizers are inspired by nature phenomena, which include genetic algorithm (GA) [25], evolution programming (EP) [26,27], evolution strategy (ES) [28,29], differential evolution (DE) [6,30], ant colony optimization (ACO) [31], particle swarm optimization (PSO) [32][33][34][35][36][37], bacterial foraging optimization (BFO) [38], simulated annealing (SA) [39], tabu search (TS) [40], harmony search (HS) [35,36,40], etc. These optimizers facilitated research into the optimization of the subproblems.…”

Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front

“…In this case, the stochastic convergence of EAs can be studied by analyzing the convergence of a random variable sequence (r.v.s.). The theoretical analyses of real-coded EAs are mainly studied in this way, as in [39,3,15,4,5,26,27]. Additionally, this method has been applied by He and Yao [28,29] to analyses of some discrete-coded EAs, and by Laummans et al [33] to successfully analyze the convergence properties of a discrete-coded MOEA.…”

Efficient covariance matrix update for variable metric evolution strategies

“…Beyer 2001;Auger 2005;Jägersküpper 2006Jägersküpper , 2008. Let the optimization problem be defined by an objective function f : R n → Y to be minimized, where n denotes the dimensionality of the search space (space of candidate solutions, decision space) and Y the space of cost values.…”