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## GRUNGE: Randomized UNcorrelated Gaussian Extrema

This is a function generator, not a single function. Hence the reason why the images look different. Different fitness landscapes result from different inputs to the generator.

## Example 1: 20 Gaussians

This landscape was generated from 20 randomly positioned gaussians with weights selected uniformly at random from the range 0 to 1 and widths selected uniformly at random in the range 0 to 10.

## Example 2: 20 Gaussians

This landscape was generated from 20 randomly positioned gaussians with weights selected uniformly at random from the range 0 to 1 and widths selected uniformly at random in the range 2 to 4.

## Example 3: 200 Gaussians

This landscape was generated from 200 randomly positioned gaussians with weights selected uniformly at random from the range 0 to 1 and widths selected uniformly at random in the range 2 to 4.

## Example 4: 200 Gaussians

This landscape was generated from 200 randomly positioned gaussians with weights selected uniformly at random from the range 0 to 1 and widths selected uniformly at random in the range 2 to 4.

## Latex

A maximization problem:

$$f(x_0 \cdots x_n) = \sum_{i=0}^M \xi_i \cdot exp[-\zeta_i \cdot \sum_{j=0}^n (x_j - \kappa_j)^2]$$

with the random numbers $\xi_i$, $\zeta_i$ and $\kappa_j$ being the weight, width and position of the $i^{\text{th}}$ Gaussian in $N$-dimensional space.

I used weights selected uniformly at random from the range 0 to 1 and widths selected uniformly at random from the range 2 to 4.

$$0 \leq x_i \leq 10$$

$$\text{maximum at }f(???) = ?$$

The maximum is not generally known. and will be different with different choices for the random values.

## Python

Due to speed issues, all the randomized gaussian landscapes were compiled using cython. Code for one 20-gaussian landscape follows:
hills0 = [[4.20571580830845, 2.5891675029296337, 5.112747213686085, 4.049341374504143, 7.837985890347726, 3.0331272607892745, 4.765969541523559, 5.833820394550312, 9.081128851953352, 5.046868558173903],[6.183689966753317, 2.5050634136244057, 9.097462559682402, 9.827854760376532, 8.102172359965895, 9.021659504395828, 3.1014756931933265, 7.298317482601286, 8.988382879679936, 6.839839319154413],[4.341718354537837, 6.1088697344380165, 9.130110532378982, 9.666063677707587, 4.77009776552717, 8.6530992777164, 2.604923103919594, 8.050278270130223, 5.486993038355893, 0.14041700164018955],[8.24844977148233, 6.681532012318509, 0.011428193144282783, 4.935778664653246, 8.676027754927809, 2.4391087688713196, 3.252043627473901, 8.704712321086546, 1.9106709150239054, 5.675107406206719],[8.0317946927987, 4.479695714355704, 0.8044581855253541, 3.2005460467254574, 5.07940642520574, 9.328338242269067, 1.0905784593110368, 5.512672460905512, 7.065614098668896, 5.474409113284238],[9.638385459738009, 6.03185627961383, 5.8761706417543635, 4.4498902627551615, 5.962868615831063, 3.8490114597266043, 5.756510141648885, 2.90329502402758, 1.8939132855435614, 1.867295282555551],[4.765309920093808, 0.8982436119559367, 7.576039219664368, 8.767703708227748, 9.233810159462806, 8.424602231401824, 8.98173121357879, 9.230824398201769, 5.405999249480544, 3.912960502346249],[8.116287085078785, 8.49485965186367, 8.950389674266752, 5.898011835311598, 9.497648732321206, 5.796950107456059, 4.505631066311552, 6.60245378622389, 9.962578393535727, 9.16941217947456],[6.1278310504071225, 4.864442019691668, 6.301473404114728, 8.450775756715153, 2.4303562206185623, 7.3148922079084775, 1.1713429320851798, 2.204605368678285, 7.9458297171057595, 3.3253614921965546],[1.4635848891230385, 6.976706401912388, 0.4523406786561235, 5.7386603678916694, 9.100160146990397, 5.341979682607239, 6.805891325622565, 0.26696794662205203, 6.349999099114583, 6.063384177542189],[3.701399403351875, 9.805166506472688, 0.36392037611485795, 0.21636509855024078, 9.61031280239611, 1.8497194139743833, 1.2389516442443171, 2.1057650988664642, 8.00746590354181, 9.369691586445807],[1.0150021937416975, 2.5991988979283196, 2.2082927131631735, 6.469257198353225, 3.502939673965323, 1.8031790152968785, 5.036365052098873, 0.39378707084692377, 1.0092124118896661, 9.88235148722501],[7.315983062253606, 8.383265651934163, 9.184820619953314, 1.6942460609746768, 6.726405635730526, 9.665489030431832, 0.5805094382649867, 6.7620178429937825, 8.454245937016164, 3.4231254107858398],[4.423140336990789, 1.7481948445144113, 4.71625415096288, 4.0990539565755455, 5.691127395242802, 5.086001300626331, 3.114460010002068, 3.571516825902629, 8.37661174368979, 2.5093266482213705],[7.415743774106636, 3.359165544734606, 0.4569649356841665, 2.8088316421834825, 2.4013040782635398, 9.531293398277988, 3.5222556151550743, 2.878779148564, 3.5920119725374633, 9.46905835657891],[7.156193503014563, 3.8801723531250563, 4.144179882772473, 6.50832862263345, 0.01524221856720187, 1.923095412446758, 3.344016906625016, 2.3941596018595854, 6.373994011293003, 3.7864807032309447],[4.144063966836443, 4.0226707511907955, 7.018296239336754, 4.18226553292466, 6.621958889738174, 0.46779685956798267, 4.453521897188298, 2.5922692344722273, 1.5768657212231085, 5.275731301676147],[7.554847672586825, 8.838751542487008, 4.945826703752868, 3.12058246416873, 4.668922353525236, 8.090458573603623, 8.75016331480271, 8.12414932363759, 1.88001294050828, 9.994203594553303],[7.255543554613125, 9.868214802051282, 4.018168222125436, 6.785150052419683, 3.1617713722134235, 2.135246620646961, 7.173241433110372, 0.023575647193538884, 8.227314105314157, 5.283459768597928],[6.492654248961536, 8.736538239003423, 2.7998274332687254, 9.785151867733981, 1.0018068906370903, 8.539381095973383, 3.9669617733090448, 0.8134541676823415, 2.747138434192621, 4.529781848179143]]
width0 = [3.515908805880605, 3.511608408314448, 2.2014024161367316, 2.7976470844485375, 3.9350805005802867, 3.0805672139406477, 3.3133187779792577, 2.5512682426242543, 2.164745976393295, 2.201215040432192, 2.7824188186456538, 2.8512376639336345, 2.7171106026232037, 3.193582786938822, 2.024872637658629, 3.2421536912373345, 3.1363028418203838, 3.122809851228854, 2.1669341003514586, 2.2378077895694917]
weight0 = [0.8444218515250481, 0.28183784439970383, 0.47214271545271336, 0.7197046864039541, 0.23861592861522019, 0.814466863291336, 0.6127731798686067, 0.7052833998544062, 0.7933250841302242, 0.8159130965336595, 0.5759529480315407, 0.022782575668658378, 0.19935579046706298, 0.25068733928511167, 0.560600218853524, 0.6337478522492526, 0.8754233917130172, 0.48726560106903205, 0.6330887599183004, 0.09778434180065931]

def fitnessFunc0(list chromosome):
cdef int num_hills = 20
cdef int dimensions = 10
cdef double subTotal, total = 0
for i in xrange(num_hills):
subTotal = 0
ith_hill = hills0[i]
for j in xrange(dimensions):
subTotal += (chromosome[j] - ith_hill[j])**2
total += weight0[i] * exp(-width0[i] * subTotal)


## 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}
}

A general-purpose tunable landscape generator
@ARTICLE{1705405,
author={Gallagher, M. and Bo Yuan},
journal={Evolutionary Computation, IEEE Transactions on},
title={A general-purpose tunable landscape generator},
year={2006},
month={oct. },
volume={10},
number={5},
pages={590 -603},
keywords={continuous optimization;distribution algorithm estimation;empirical algorithm;evolutionary computation;general-purpose tunable landscape generator;metaheuristic;test-problem;algorithm theory;distributed algorithms;evolutionary computation;},
doi={10.1109/TEVC.2005.863628},
ISSN={1089-778X},}

## Notes

More example landscapes follow:

## Example 6: 200 Gaussians

The above examples were generated with weights randomly selected in the range 0 to 1 and widths randomly selected in the range 2 to 4.

## Example 4: 200 Gaussians

The above examples were generated with weights randomly selected in the range 0 to 1 and widths randomly selected in the range 1 to 2.

## Example 7: 200 Gaussians

The above examples were generated with weights randomly selected in the range 0 to 1 and widths randomly selected in the range 0 to 10.

## Example 3: 200 Gaussians

The above examples were generated with weights randomly selected in the range 0 to 1 and widths randomly selected in the range 0 to 1.