Adaptation in natural and artificial systems
Adaptation in natural and artificial systems
The theory of evolution strategies
The theory of evolution strategies
Evolution and Optimum Seeking: The Sixth Generation
Evolution and Optimum Seeking: The Sixth Generation
On the analysis of the (1+ 1) evolutionary algorithm
Theoretical Computer Science
Proceedings of the 6th International Conference on Genetic Algorithms
An Analysis Of The Role Of Offspring Population Size In EAs
GECCO '02 Proceedings of the Genetic and Evolutionary Computation Conference
Evolutionary Algorithms and the Maximum Matching Problem
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Rigorous runtime analysis of a (μ+1)ES for the sphere function
GECCO '05 Proceedings of the 7th annual conference on Genetic and evolutionary computation
Real royal road functions for constant population size
GECCO'03 Proceedings of the 2003 international conference on Genetic and evolutionary computation: PartII
Analysis of a simple evolutionary algorithm for minimization in euclidean spaces
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Worst-case and average-case approximations by simple randomized search heuristics
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Rigorous runtime analysis of the (1+1) ES: 1/5-rule and ellipsoidal fitness landscapes
FOGA'05 Proceedings of the 8th international conference on Foundations of Genetic Algorithms
Evolutionary programming made faster
IEEE Transactions on Evolutionary Computation
Population size versus runtime of a simple evolutionary algorithm
Theoretical Computer Science
Empirical investigation of simplified step-size control in metaheuristics with a view to theory
WEA'08 Proceedings of the 7th international conference on Experimental algorithms
How comma selection helps with the escape from local optima
PPSN'06 Proceedings of the 9th international conference on Parallel Problem Solving from Nature
PPSN'12 Proceedings of the 12th international conference on Parallel Problem Solving from Nature - Volume Part I
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We consider the (1+λ)ES and the (1+λ)ES, which are simple evolutionary algorithms for minimization in Rn, using isotropic mutations. General lower bounds on the number of mutations that are necessary to reduce the approximation error in the search space, ie the distance from the optimum (or from any other fixed point in the search space), are proved. Therefore, we generalize a lower-bound method recently introduced by Witt in a runtime analysis of the (μ+1)EA for the search space {0,1}n, which was also already successfully applied in an analysis of a (μ+1)ES. Namely, we prove that both, the (1+λ)ES as well as the (1+λ)ES need Ω(n•λ/√lnλ) function evaluations with an overwhelming probability to halve the approximation error in the search space - independently of how the isotropic mutations are adapted and of the function to be optimized.On the other hand, for an upper bound we consider the following concrete scenario: the minimization of the well-known SPHERE-function using Gaussian mutation vectors adapted by the 1/5-rule. We prove that the (1+λ)ES needs Ω(n•λ/√lnλ). SPHERE-evaluations with an overwhelming probability to halve the approximation error. Moreover, by some kind of reduction, we show that this upper bound also holds for the (1,λ)ES.Finally, the gap of size O(√lnλ) between the lower bound and the upper bound is discussed.