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Schaffer's F6

Two dimensional view

One dimensional view

Fine grain view

Function

Latex

A minimization problem:

$$f(x,y)=0.5+\frac{sin^2(\sqrt{x^2 + y^2})-0.5}{[1+0.001 \cdot (x^2 + y^2)]^2}$$

$$-100 \leq x_i \leq 100$$

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

In the python code below I use the weighted wrap method for expanding functions. For example, expanding the two dimensional function $f(x,y)$ to four dimensions: $g(x_1,x_2,x_3,x_4)=w_1f(x_1,x_2)+w_2f(x_2,x_3)+w_3f(x_3,x_4)+w_4f(x_4,x_1)$. This is described in more detail in "The Island Model Genetic Algorithm: On Separability, Population Size and Convergence".

Python

def fitnessFunc(self, chromosome):
	""" """
	if len(chromosome) <= 2:
		return self.schafferF6HelperFunction(0, chromosome)
	else:
		total = 0
		for i in xrange(len(chromosome)):
			total += self.weights[i] * \
			self.schafferF6HelperFunction(i, chromosome)
		return total

def schafferF6HelperFunction(self, index, chromosome):
	x = chromosome[index]
	y = chromosome[(index+1)%len(chromosome)]
	xsqrdysqrd = x**2 + y**2
	return 0.5 + (math.sin(math.sqrt(xsqrdysqrd))-0.5) / (1+0.001*xsqrdysqrd)**2

Sources

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

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}
}

The Island Model Genetic Algorithm: On Separability, Population Size and Convergence
@article{separabilityConvergence,
	location = {http://www.scientificcommons.org/42876414},
	title = {The Island Model Genetic Algorithm: On Separability, Population Size and Convergence},
	author = {Darrell Whitley and Soraya Rana and Robert B. Heckendorn},
	abstract = {Parallel Genetic Algorithms have often been reported to yield better performance than Genetic Algorithms which use a single large panmictic population. In the case of the Island Model genetic algorithm, it has been informally argued that having multiple subpopulations helps to preserve genetic diversity, since each island can potentially follow a different search trajectory through the search space. It is also possible that since linearly separable problems are often used to test Genetic Algorithms, that Island Models may simply be particularly well suited to exploiting the separable nature of the test problems. We explore this possibility by using the infinite population models of simple genetic algorithms to study how Island Models can track multiple search trajectories. We also introduce a simple model for better understanding when Island Model genetic algorithms may have an advantage when processing some test problems. We provide empirical results for both linearly separa...},
	journal = {Journal of Computing and Information Technology},
    year = {1998},
    volume = {7},
    pages = {33--47},
	url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=?doi=10.1.1.36.7225},
	institution = {CiteSeerX - Scientific Literature Digital Library and Search Engine [http://citeseerx.ist.psu.edu/oai2] (United States)},
}


Notes