Epistatic Michalewicz Problem $($EM$)($second ICEO$)$

$mi{n}_{x}f(x)=-{\displaystyle \underset{i=1}{\overset{n}{}}}sin(y_i)(sin(i\frac{y_i^2}{}){)}^{2m}$

$y_i=\{\begin{array}{ccc}{x}_{i}cos()-x_{i+1}sin(),& i=1,3,5,\text{d}\text{o}\text{t}\text{s}\text{,}& n\end{array}$

$x_{i}sin()+x_{i+1}cos(),i=2,4,6,$dots,$n$

$x_i,i=n$

subject to $0x_i,$iin$\{1,2,$dots,$n\},=\frac{}{6}$ and $m=10.$ The number of local minima is not known but the global minimizer is presented in Table $3.$

Table $3.$ Epistatic Michalewicz's global optimizers

$\backslash$table$[[n,f(x^{**}),$

Main Task

Write a MATLAB program to find the minimum of the function $f_{cd}(*)$ where $cd=ab(mod50)$ from $,$ Appendix B$]$ by $.$using

Newton$-$Raphson,

Hestenes$-$Stiefel,

Polak$-$Ribi$$re and

Fletcher$-$Reeves algorithms.

You are also strongly encouraged to use another relevant algorithm from the literature, which will be rewarded with an extra $10$ points. If the function is not differentiable, use an approximation proposed by yourself or an relevant approximation commonly used in the literature and write it explicitly in your report. Repeat the main steps of your algorithms until the desired accuracy is achieved, i$.$e$.$

$||gradf(x_k)||$ and $|f(x_{k+1})-f(x_k)|$

Take THREE initial guess as $x_0{N}_n(0,1)(n-$dimensional vector having elements from standard normal distribution by using randn function of MATLAB$)$ or $-$dimensional vector having elements from uniform distribution from the closed interval $x_{0,\text{m}\text{i}\text{n}},x_{0,\text{m}\text{a}\text{x}}$ where $x_{0,\text{m}\text{i}\text{n}}$ and $x_{0,\text{m}\text{a}\text{x}}$ are specified for each function in $[1]$ by using rand function of MATLAB$).$ For instance, if your problem is defined on $-2x_{i}2,$ for $i=1,2,$ you may consider to choose $x_0{U}_n(-2,2).$ Take also the absolute error bound as $={10}^{{?}^{-4}}$ for every algorithm.$)$