Drift analysis and linear functions revisited

Doerr, Benjamin; Johannsen, Daniel; Winzen, Carola

doi:10.1109/cec.2010.5586097

Cited by 40 publications

(54 citation statements)

References 10 publications

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“…Using the theorem, we prove that the OneMax and the Trap problem are the easiest and the hardest function in the pseudo-Boolean function class with a unique global optimum for the (1+1)-EA with mutation probability less than 0.5, respectively, which much extends the previous knowledge [3]. In the future, we will apply this theorem for more problem classes and more algorithms.…”

Section: Repeat Until the Termination Condition Is Met 3 Xmentioning

confidence: 55%

“…However, these studies did not concern the boundary problem cases. Recently, Doerr et al [3] used the lower bound of running time of the (1+1)-EA with mutation probability 1 n on the OneMax problem as that on the pseudo-Boolean function class with a unique global optimum by proving that the OneMax problem is the easiest case for the (1+1)-EA in this class, comparing to which this paper derives a more general result for the easiest case as well as the hardest case. Note that, the easiest case derived in this paper has also been proved by Witt [13], but we give a different and more compact proof.…”

Section: Introductionmentioning

confidence: 97%

“…The running time of EAs on linear pseudo-Boolean function class, which is a relatively small problem class, was analyzed in [5,8,9,3]. Yu and Zhou [17] provided a general idea on why EAs can fail over a complex problem class; Fournier and Teytaud [6] provided a general lower bound for the performance of EAs over problem classes with VC-dimension measured complexity.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Algorithm-Dependent Boundary Case Identification for Problem Classes

Qian

Zhou

2012

Lecture Notes in Computer Science

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Abstract. Running time analysis of metaheuristic search algorithms has attracted a lot of attention. When studying a metaheuristic algorithm over a problem class, a natural question is what are the easiest and the hardest cases of the problem class. The answer can be helpful for simplifying the analysis of an algorithm over a problem class as well as understanding the strength and weakness of an algorithm. This algorithm-dependent boundary case identification problem is investigated in this paper. We derive a general theorem for the identification, and apply it to a case that the (1+1)-EA with mutation probability less than 0.5 is used over the problem class of pseudo-Boolean functions with a unique global optimum.

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Section: Repeat Until the Termination Condition Is Met 3 Xmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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On Algorithm-Dependent Boundary Case Identification for Problem Classes

Qian

Zhou

2012

Lecture Notes in Computer Science

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“…Based on an idea by Doerr, Johannsen and Winzen [1], Sudholt [18] showed that among all functions with a unique global optimum OneMax is the easiest function. This holds with respect to a broad class of mutation-based EAs, which includes the (1,λ) EA.…”

Section: A Sharp Threshold For the Off-spring Population Sizementioning

confidence: 99%

The choice of the offspring population size in the (1,λ) EA

Rowe

Sudholt

2012

Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation

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We extend the theory of non-elitist evolutionary algorithms (EAs) by considering the offspring population size in the (1,λ) EA. We establish a sharp threshold at λ = log e e−1 n ≈ 5 log 10 n between exponential and polynomial running times on OneMax. For any smaller value, the (1,λ) EA needs exponential time on every function that has only one global optimum. We also consider arbitrary unimodal functions and show that the threshold can shift towards larger offspring population sizes. Finally, we investigate the relationship between the offspring population size and arbitrary mutation rates on OneMax. We get sharp thresholds for λ that decrease with the mutation rate. This illustrates the balance between selection and mutation.

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“…For a simple elitist (1+1) evolutionary algorithm (EA) using only standard bit mutations (SBM) Doerr, Johannsen, and Winzen showed that OneMax, a function simply counting the number of ones in a bit string, is the easiest problem among all functions with a single global optimum [8]. This statement generalises to the class of all evolutionary algorithms that only use standard bit mutation for variation [33] as well as higher mutation rates and stochastic dominance [37].…”

Section: Introductionmentioning

confidence: 99%

On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation

Corus

Jansen

et al. 2016

Algorithmica

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Understanding which function classes are easy and which are hard for a given algorithm is a fundamental question for the analysis and design of bio-inspired search heuristics. A natural starting point is to consider the easiest and hardest functions for an algorithm. For the (1+1) EA using standard bit mutation (SBM) it is well known that OneMax is an easiest function with unique optimum while Trap is a hardest. In this paper we extend the analysis of easiest function classes to the contiguous somatic hypermutation (CHM) operator used in artificial immune systems. We define a function MinBlocks and prove that it is an easiest function for the (1+1) EA using CHM, presenting both a runtime and a fixed budget analysis. Since MinBlocks is, up to a factor of 2, a hardest function for standard bit mutations, we consider the effects of combining both operators into a hybrid algorithm. We rigorously prove that by combining the advantages of k operators, several hybrid algorithmic schemes have optimal asymptotic performance on the easiest functions for each individual operator. In particular, the hybrid algorithms using CHM and SBM have optimal asymptotic performance on both OneMax and MinBlocks. We then investigate easiest functions for hybrid schemes and show that an easiest function for a hybrid algorithm is not just a trivial weighted combination of the respective easiest functions for each operator.

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Drift analysis and linear functions revisited

Cited by 40 publications

References 10 publications

On Algorithm-Dependent Boundary Case Identification for Problem Classes

On Algorithm-Dependent Boundary Case Identification for Problem Classes

The choice of the offspring population size in the (1,λ) EA

On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation

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