New evolutionary optimization method based on information sets

Abstract

This paper proposes a new evolutionary learning method without any algorithmic-specific parameters for solving optimization problems. The proposed method gets inspired from the information set concept that seeks to represent the uncertainty in an effort using an entropy function. This method termed as Human Effort For Achieving Goals (HEFAG) comprises two phases: Emulation and boosting phases. In the Emulation phase the outcome of the best achiever is emulated by each contender. The effort associated with the average outcome and best outcome are converted into information values based on the information set. In the Boosting phase the efforts of all contenders are boosted by adding the differential information values of any two randomly chosen contenders. The proposed method is tested on benchmark standard functions and it is found to outperform some well-known evolutionary methods based on the statistical analysis of the experimental results using the Kruskal-Wallis statistical test and Wilcoxon rank sum test.

Appendix A: Some standard functions

The functions tested in the paper are as follows:

• 1. Sphere function $f_{1}(x)={\sum }_{i = 1}^{n}{x_{i}^{2}}$ has global minimum $x_{i}^{*}= 0$ and minimum function value f(x^∗) = 0.

• 2. Rosenbrock function $f_{2}(x)={\sum }_{i = 1}^{n}(x_{i + 1}-x_{i})^{2}+(x_{i}-1)^{2}$ has global minimum $x_{i}^{*}= 0$ and minimum function value f(x^∗) = 0.

• 3. Sum of different power funcions $f_{3}(x)={\sum }_{i = 1}^{n}|x_{i}^{i + 1}|$ has global minimum $x_{i}^{*}= 0$ and minimum function value f(x^∗) = 0.

• 4. Schwefel26 function $f_{4}(x)=x_{i}sin(\sqrt {x_{i}})$ has global minimum $x_{i}^{*}= 0$ and minimum function value f(x^∗) = 0.

• 5. Rastrigin function $f_{5}(x)={\sum }_{i = 1}^{n}(x_{i}-10cos(2\pi x_{i})+ 10)$ has global minimum $x_{i}^{*}= 0$ and minimum function value as f(x^∗) = 0.

• 6. Griewank function $f_{6}(x)=\frac {1}{4000}{\sum }_{i = 1}^{n}{x_{i}^{2}}-{\prod }_{i = 1}^{n}cos(\frac {x_{i}}{\sqrt {i}})+ 1$ has global minimum $x_{i}^{*}= 0$ and minimum function value as f(x^∗) = 0.

• 7. Qing function $f_{7}(x)={\sum }_{i = 1}^{n}(x_{i}-i)^{2}$ has global minimum $x_{i}^{*}=\sqrt {i}$ and minimum function value f(x^∗) = 0.

• 8. Alpine01 function f₈(x) = |x_isin(x_i) + 0.1x_i| has global minimum $x_{i}^{*}= 0$ and minimum function value as f(x^∗) = 0.

• 9. Ackleley function $f_{9}(x)= -20 e^{(-0.2 \sqrt {\frac {1}{n} {\sum }_{i = 1}^{n} {x_{i}^{2}}})} - e^{(-\frac {1}{n} {\sum }_{i = 1}^{n} cos(2\pi x_{i}))} + 20 + e$ has global minimum $x_{i}^{*}= 0$ and minimum function value as f(x^∗) = 0.

• 10. Zakharov function $f_{10}(x)={\sum }_{i = 1}^{n} {x_{i}^{2}}+({\sum }_{i = 1}^{n} 0.5ix_{i})^{2} + ({\sum }_{i = 1}^{n} 0.5ix_{i})^{4}$ has global minimum $x_{i}^{*}= 0$ and minimum function value as f(x^∗) = 0.