RGA - algorithm for black box global optimization

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RGA  in classic style.

Six-hump camel back function

Fig. 4.   Six-hump camel back function [1].  Within the bounded  region six local minima are located,  two of them  are global minima.

Function definition
f (x,y) = ( 4 - 2.1 x² +  x⁴/3) x² + xy + ( - 4  +  4y²) y²;    
     -3 <= x <= 3, -2 <= y <= 2.
Global minimum
    f (x, y) =  - 1.0316;  (x, y)  =  (-0.0898,0.7126), (0.0898,-0.7126).

NB! Please, pay attention: isolines are not equidistant, but strongly dense near  '0' .

 The results of two RGA-runs are:
  I)   (x, y)  =  ( 0.089849, -0.712656) ;  f (x,y) =  -1.0316280
II)   (x, y)  = (-0.088004,   0.711162) ;  f (x,y) =  -1.0315998
The number of OF calls are 1023 and 931 respectively.

n - dimensional
Rosenbrock's valley (Banana function)

Rosenbrock's valley [2] , also known as Banana function, is a classic optimization problem. The global optimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial, however convergence to the global optimum is difficult and hence this problem has been repeatedly used in assess the performance of optimization algorithms.

Function definition
    f (x) =  sum_i  [ 100 ( x_i+1 - x_i ² ) ²  + ( 1 - x_i ) ²  ],    i = 1 : n-1;
   -2.048 <=  x_i <= 2.048.
Global minimum
    f (x) = 0;   x_i = 1,   i = 1 : n.

 The results of RGA-runs.

n = 2

Fig. 5.  Banana function in 2-dimensional space.
NB!  Isolines are not equidistant,  but  of  logarithmic scale .

The results of  RGA-run are:
x =  (0.99882,  0.99622); f (x)  =   0.000202
Number of OF  calls is 428.

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  n = 4

The results of  threee RGA-runs are ( 3 realizations of a starting set: 71 random points)
I.      x =  (1.0001,  1.0036,  1.0071,   1.0191);     f (x)  =   0.0028649
II.    x =  (1.0000,  0.9989,   0.9971,   0.9959);    f (x)  =  0.0004630
III.  x =  (1.0006,  1.0011,   1.0002,   1.0017);    f (x)  =   0.0006531
Numbers of OF  calls  are  1942,  1418,  1714..

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n = 5 The result of  RGA-run is:
x =  (1.0028,  1.0057,  1.0221,  1.0102,  1.0422);     f (x)  =   0.0015604
Number of OF  calls is 2564.

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n = 6 The result of  RGA-run is:
x =  (0.9996,  0.9994,   0.9990,   0.9984,   0.9969,  0.9940);  f (x)  = 0.0000339
Number of OF  calls is 4794.

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n = 7 The result of  RGA-run is:
x =  (1.0000,  1.0000,  1.0000, 1.0000, 1.0000,  1.0001, 1.0002);  f (x)  = 0.00000023
Number of OF  calls is 5531.

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n = 8 The result of  RGA-run is:
x =  (0.9996,  0.9993,   0.9993,   0.9985,   0.9971,  0.9940,  0.9878,  0.9759 );  f (x)  = 0.00022791
Number of OF  calls is 22081

[1] Dixon, L. C. W. and Szego, G. P.: The optimization problem: An introduction. in Dixon, L. C. W. and Szego, G. P. (Eds.), Towards Global Optimization II, New York: North Holland, 1978.
[2] Rosenbrock H. H., "An Automatic Method for Finding the Greatest or Least Value of a Function," Computer Journal, Vol 3, pp. 175-184, 1960.

I am pleased to thank  Robert B. Love and  Evgueni Petrov: while testing their  own software  they found  the  error in printing  the  Six-hump camel back function - the very last  term of  the  Camel  should  definitely look like  "y²" as it  looks now ,  but  not  "x²"  as it was prior to 24.05.2000 .
As  to  the  correct  definition  of another test-functions,  please  find,  e.g.,  the GlobOpt-page  

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