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Griewank

Two dimensional view

One dimensional view

Fine grain view

Function

Latex

A minimization problem:

$$f(x_1 \cdots x_n) = 1 + \frac{1}{4000} \sum_{i=1}^n x^2_i - \prod_{i=1}^n cos(\frac{x_i}{\sqrt{i}})$$

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

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

Python

def fitnessFunc(self, chromosome):
	"""F6 Griewank's function
	multimodal, symmetric, inseparable"""
	part1 = 0
	for i in range(len(chromosome)):
		part1 += chromosome[i]**2
		part2 = 1
	for i in range(len(chromosome)):
		part2 *= math.cos(float(chromosome[i]) / math.sqrt(i+1))
	return 1 + (float(part1)/4000.0) - float(part2)

Sources

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

The following paper by Jumonji, et al. has multiple typos in Table 1 including a missing cosine for F6 Griewank's function. The function is written correctly elsewhere in the paper.
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},
}

The following paper by Whitley, et al. shows slices of Griewank's function for 1, 3, 5, and 10 dimensional versions of this problem on page 251. The caption states that as the dimensionality increases, the local optima induced by the cosine decrease in number and complexity. I was not able to replicate this result using my visualization.py script's -slice flag. Perhaps Whitley, et al. are graphing only the product term that includes the cosine? whereas I was slicing through Griewank's function in full.
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 F8 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