Optimization-Project 1 Solved

Eighteen selected problems are standard benchmark functions of different properties: Schwefel, De Jong 1, Rosenbrock’s Saddle, Rastrigin, Griewangk, Sine Envelope Sine Wave, Stretch V Sine Wave, Ackley One, Ackley Two, Egg Holder, Rana, Pathological, Michalewicz, Master’s Cosine Wave, Quartic, Levy, Step and Alpine. All of the functions are dimension-wise scalable.

Table 1 presents functions together with optimal values, in cases where global optima is known and can be reasonably expressed independent of dimension. The third column gives dimensions used in the experimentation for each function. The last column is the search and initialization range used in the experimentation.

1.    Schwefel’s function:

                                                                     (1)

2.    1st De Jong’s function:

n

                                                                                            f2 (x) = Xx2i                                                                     (2)

i=1

3.    Rosenbrock

                                                                                  (3)

4.    Rastrigin

                                                                                 (4)

5.    Griewangk

                                                                                       (5)

6.    Sine Envelope Sine Wave

                                                                        (6)

7.    Stretched V Sine Wave

!

(7) 8. Ackley’s One

                                            ))     (8)

9.       Ackley’s Two

                                          (9)

10.   Egg Holder

                                             (10)

11.   Rana

                                                                                                               (11)

12.   Pathological

                                                    (12)

13.   Michalewicz

                                                                           (13)

14.   Masters Cosine Wave

                                                          (14)

15.   Quartic

           (15) 16. Levy



where: 17. Step

n−1

                                                                                  f17 (x) = X(|xi| + 0.5)2                                                   (17)

i=0

18. Alpine

n−1

                                                                         f18 (x) = X|xi · sin(xi) + 0.1 · xi|                              (18)

i=0

Table 1: Experiments



f1 Schwefel    0                  10,20,30    [ f2                      De Jong 1  0   10,20,30    [ f3                      Rosenbrock’s Saddle  0                  10,20,30    [ f4     Rastrigin   0                  10,20,30    [ f5                      Griewangk                    0   10,20,30    [ f6                      −1.4915(n − 1)            10,20,30    [ f7     Stretch V Sine Wave  0                  10,20,30    [ f8                      −7.54276 − 2.91867(n − 3)               10,20,30    [ f9                      Ackley Two                  0   10,20,30    [

f10
Egg Holder
−
10,20,30
[−500,500]n
f11
Rana
−
10,20,30
[−500,500]n
f12
Pathological
−
10,20,30
[−100,100]n
f13
Michalewicz
0.966n
10,20,30
[0,π]n
f14
Masters’ Cosine Wave
1 − n
10,20,30
[−30,30]n
f15
Quartic
0
10,20,30
[−100,100]n
f16
Levy
0
10,20,30
[−10,100]n
f17
Step
0
10,20,30
[−100,100]n
f18
Alpine
0
10,20,30
[−100,100]n


Pseudo-random number generator
Use the Mersenne Twister (MT) pseudo-random number generator in your code. The MT webpage is at (http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html) and the different programming language codes are available at (http://www.math.sci.hiroshima-u. ac.jp/~m-mat/MT/VERSIONS/eversions.html).

Experiment
Generate at least 30 pseudo-random solution vectors and solve for all functions in given dimensions of 10, 20 and 30. Compute statistical analysis on the obtained results for average, standard deviation, range, median and time (in millisecond).