Eighteen selected problems are standard benchmark functions of different properties: Schwefel, De Jong 1, Rosenbrocks Saddle, Rastrigin, Griewangk, Sine Envelope Sine Wave, Stretch V Sine Wave, Ackley One, Ackley Two, Egg Holder, Rana, Pathological, Michalewicz, Masters 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. Schwefels function:

(1)

2. 1st De Jongs 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. Ackleys One

)) (8)

9. Ackleys 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

n1

f₁₇ (x) = ^X(|x_i| + 0.5)² (17)

i=0

18. Alpine

n1

f₁₈ (x) = ^X|x_i sin(x_i) + 0.1 x_i| (18)

i=0

Table 1: Experiments

f₁ Schwefel 0 10,20,30 [ f₂ De Jong 1 0 10,20,30 [ f₃ Rosenbrocks 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 [

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).

Submission

The student must submit the following separate files to canvas:

1. source codes
2. a L^ATEX typeset report on the results and its analysis

The report must contain an introduction in the problems, the full experimentation results in tabular format and condensed results with statistical analysis.

The files must be submitted through Canvas by 5PM April 5, 2019. The grading rubric is given in Table 2.

Table 2: Grading rubric