Home

Schwefel

Two dimensional view

One dimensional view

Function

Latex

A minimization problem:

$$f(x_1 \cdots x_n) = \sum_{i=1}^n (-x_i sin(\sqrt{|x_i|})) + \alpha \cdot n$$

$$\alpha = 418.982887$$

$$-512 \leq x_i \leq 512$$

$$\text{minimum at }f(420.968746, 420.968746, \cdots, 420.968746) = 0$$

Python

def fitnessFunc(self, chromosome):
	"""F7 Schwefel's function
	multimodal, asymmetric, separable"""
	alpha = 418.982887
	fitness = 0
	for i in range(len(chromosome)):
		fitness -= chromosome[i]*math.sin(math.sqrt(math.fabs(chromosome[i])))
	return float(fitness) + alpha*len(chromosome)

Sources

The following may or may not contain the originator of this function.

www-optima.amp.i.kyoto-u.ac.jp

A novel distributed genetic algorithm implementation with variable number of islands
@inproceedings{varIslandNum07,
author = {Takuma Jumonji and Goutam Chakraborty and Hiroshi Mabuchi and Masafumi Matsuhara},
title = {A novel distributed genetic algorithm implementation with variable number of islands},
booktitle = {IEEE Congress on Evolutionary Computation},
year = {2007},
pages = {4698--4705},
doi = {10.1109/CEC.2007.4425088},
masid = {4737000}
}

The influence of migration sizes and intervals on island models
@inproceedings{Skolicki:2005:IMS:1068009.1068219,
 author = {Skolicki, Zbigniew and De Jong, Kenneth},
 title = {The influence of migration sizes and intervals on island models},
 booktitle = {Proceedings of the 2005 conference on Genetic and evolutionary computation},
 series = {GECCO '05},
 year = {2005},
 isbn = {1-59593-010-8},
 location = {Washington DC, USA},
 pages = {1295--1302},
 numpages = {8},
 url = {http://doi.acm.org/10.1145/1068009.1068219},
 doi = {10.1145/1068009.1068219},
 acmid = {1068219},
 publisher = {ACM},
 address = {New York, NY, USA},
 keywords = {evolutionary computation, island model, migration, migration interval, migration size, test suite},
}

Empirical review of standard benchmark functions using evolutionary global optimization
@article{DBLP:journals/corr/abs-1207-4318,
  author    = {Johannes M. Dieterich and
               Bernd Hartke},
  title     = {Empirical review of standard benchmark functions using evolutionary
               global optimization},
  journal   = {CoRR},
  volume    = {abs/1207.4318},
  year      = {2012},
  ee        = {http://arxiv.org/abs/1207.4318},
  bibsource = {DBLP, http://dblp.uni-trier.de}
}

Numerical Optimization of Computer Models
@book{Schwefel:1981:NOC:539468,
 author = {Schwefel, Hans-Paul},
 title = {Numerical Optimization of Computer Models},
 year = {1981},
 isbn = {0471099880},
 publisher = {John Wiley \& Sons, Inc.},
 address = {New York, NY, USA},
}

Evaluating evolutionary algorithms
@article{Whitley1996245,
title = "Evaluating evolutionary algorithms",
journal = "Artificial Intelligence",
volume = "85",
number = "1-2",
pages = "245 - 276",
year = "1996",
note = "",
issn = "0004-3702",
doi = "10.1016/0004-3702(95)00124-7",
url = "http://www.sciencedirect.com/science/article/pii/0004370295001247",
author = "Darrell Whitley and Soraya Rana and John Dzubera and Keith E. Mathias"
}

This benchmark is listed as F7 in "The parallel genetic algorithm as function optimizer"
@article{Muhlenbein1991619,
title = "The parallel genetic algorithm as function optimizer",
journal = "Parallel Computing",
volume = "17",
number = "6-7",
pages = "619 - 632",
year = "1991",
note = "",
issn = "0167-8191",
doi = "10.1016/S0167-8191(05)80052-3",
url = "http://www.sciencedirect.com/science/article/pii/S0167819105800523",
author = "H. M{\"u}hlenbein and M. Schomisch and J. Born",
keywords = "Search methods",
keywords = "optimization methods",
keywords = "parallel genetic algorithm",
keywords = "performance evaluation",
keywords = "speedup results"
}


Notes