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Lunacek

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Function

Latex

A minimization problem:

$$f(x_0 \cdots x_n) = min(\{\sum_i^N(x_i-\mu_1)^2\}, \{d \cdot N + s \cdot \sum_i^N (x_i - \mu_2)^2\})+10 \sum_i^N (1-cos 2 \pi (x_i - \mu_1))$$

$$\mu_1 = 2.5$$ (According to "Real-Parameter Black-Box Optimization Benchmarking 2009" which uses $\mu_0$ and $\mu_1$ instead of $\mu_1$ and $\mu_2$).

$$\mu_2 = - \sqrt{\frac{\mu_1^2 - d}{s}}$$

$$N = \text{number of dimensions}$$

$$d = 1$$ (According to "Real-Parameter Black-Box Optimization Benchmarking 2009")

$$s = 1 - \frac{1}{2 \sqrt{N + 20} - 8.2}$$ (According to "Real-Parameter Black-Box Optimization Benchmarking 2009")

$$-5 \leq x_i \leq 5$$

minimum at ???

Python

self.s = 1.0 - (1.0 / (2.0 * math.sqrt(self.problemDimensions + 20.0) - 8.2)) #(According to page 15 of benchmarkingFunctionsNoiseless.pdf)
self.d = 1.0 #(According to page 15 of benchmarkingFunctionsNoiseless.pdf)
self.mu1 = 2.5 #(According to page 15 of benchmarkingFunctionsNoiseless.pdf which uses \mu_0 and \mu_1 instead of \mu_1 and \mu_2)
#The abs on the next line should only be needed for 2 dimensional viewing (where the second dimension is fitness). s is only negative when self.problemDimensions==1.
self.mu2 = - math.sqrt(abs((self.mu1**2 - self.d) / self.s))

def fitnessFunc(self, chromosome):
	""""""
	firstSum = 0.0
	secondSum = 0.0
	thirdSum = 0.0
	for i in xrange(self.problemDimensions):
		firstSum += (chromosome[i]-self.mu1)**2
		secondSum += (chromosome[i]-self.mu2)**2
		thirdSum += 1.0 - math.cos(2*math.pi*(chromosome[i]-self.mu1))
	return min(firstSum, self.d*self.problemDimensions + self.s*secondSum)+10*thirdSum

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}
}

Real-Parameter Black-Box Optimization Benchmarking 2009: Noiseless Functions Definitions
@techreport{hansen:inria-00362633,
    hal_id = {inria-00362633},
    url = {http://hal.inria.fr/inria-00362633},
    title = {{Real-Parameter Black-Box Optimization Benchmarking 2009: Noiseless Functions Definitions}},
    author = {Hansen, Nikolaus and Finck, Steffen and Ros, Raymond and Auger, Anne},
    abstract = {{Quantifying and comparing performance of optimization algorithms is one important aspect of research in search and optimization. However, this task turns out to be tedious and difficult to realize even in the single-objective case -- at least if one is willing to accomplish it in a scientifically decent and rigorous way. The BBOB 2009 workshop will furnish most of this tedious task for its participants: (1) choice and implementation of a well-motivated real-parameter benchmark function testbed, (2) design of an experimental set-up, (3) generation of data output for (4) post-processing and presentation of the results in graphs and tables. What remains to be done for the participants is to allocate CPU-time, run their favorite black-box real-parameter optimizer in a few dimensions a few hundreds of times and execute the provided post-processing script afterwards. In this report, the testbed of noise-free functions is defined and motivated.}},
    language = {Anglais},
    affiliation = {TAO - INRIA Saclay - Ile de France , Microsoft Research - Inria Joint Centre - MSR - INRIA , The Process- and Product-Engineering research centre , Laboratoire de Recherche en Informatique - LRI},
    type = {Rapport de recherche},
    institution = {INRIA},
    number = {RR-6829},
    year = {2009},
    pdf = {http://hal.inria.fr/inria-00362633/PDF/RR-6829.pdf},
}

The Impact of Global Structure on Search
@incollection{springerlink:10.1007/978-3-540-87700-4_50,
   author = {Lunacek, Monte and Whitley, Darrell and Sutton, Andrew},
   affiliation = {Colorado State University Fort Collins, Colorado 80523 USA},
   title = {The Impact of Global Structure on Search},
   booktitle = {Parallel Problem Solving from Nature},
   series = {Lecture Notes in Computer Science},
   editor = {Rudolph, Gunter and Jansen, Thomas and Lucas, Simon and Poloni, Carlo and Beume, Nicola},
   publisher = {Springer Berlin / Heidelberg},
   isbn = {978-3-540-87699-1},
   keyword = {Computer Science},
   pages = {498-507},
   volume = {5199},
   url = {http://dx.doi.org/10.1007/978-3-540-87700-4_50},
   note = {10.1007/978-3-540-87700-4_50},
   year = {2008}
}


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Notes

Apparently a very difficult function, but the authors of "Empirical review of standard benchmark functions using evolutionary global optimization" argue that it is not difficult for the same reasons as real world problems. Lunacek's function, also known as the bi- or double-Rastrigin function, is a hybrid function consisting of a Rastrigin and a double-sphere part and is designed to model the double-funnel character of some difficult LJ cases. - "Empirical review of standard benchmark functions using evolutionary global optimization", citing "The Impact of Global Structure on Search"