s$f_{175}(x)=0\mathrm{.}25x_1^4-0\mathrm{.}5x_1^2+0\mathrm{.}1x_1+0\mathrm{.}5x_2^2$

subject to $-2x_{i}2,$ for $i=1,2,$ by using Newton$-$Raphson and Hestenes$-$Stiefel algorithms.

Main Task

Write a MATLAB program to find the minimum of the function $f_{cd}(*)$ where $cd=18$ 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$.$

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

Take THREE initial guess as $x_0{N}_n(0,1)(n-$dimensional vector having elements from standardubject to xi for i $=1,2,$ by using Newton$-$Raphson and Hestenes$-$Stiefel algorithms.

Main Task

Write a MATLAB program to find the minimum of the function fcd$()$ where cd $=$ ab $($mod$50)$ from

$[2,$ Appendix B$]$ by using

$$ Newton$-$Raphson,

$$ Hestenes$-$Stiefel,

$$ Polak$-$Ribi$`$ere 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$.$

$\backslash |$f$($xk and $|$f$($xk$+1)$ f$($xk

Take THREE initial guess as x$0$ Nn$(0,1)($n$-$dimensional vector having elements from standard

normal distribution by using randn function of MATLAB$)$ or x$0$ Un$($x$0,$min, x$0,$max$)($n$-$dimensional

vector having elements from uniform distribution from the closed interval $[$x$0,$min, x$0,$max$]$ where

x$0,$min and x$0,$max are specified for each function in $[1]$ by using rand function of MATLAB$).$ For

instance, if your problem is defined on xi for i $=1,2,$ you may consider to choose

x$0$ Un$(2,2).$ Take also the absolute error bound as $=104$

for every algorithm.

$1$

Instructions

Write a project report regarding to the given task by answering the following questions $($please EXPLAIN all of them by at least two sentences$)$:

$[3$p$]$ How many steps does it take to find the minimum of this function with all of these algorithms?

$[3$p$]$ What are the execution times of these algorithms? Does this make sense?

$[3$p$]$ Does the convergence depend on the initial conditions? Why?

$[3$p$]$ Based on the last two questions, what can be the reason for this trade$-$off?

$[3$p$]$ Do you expect the same number of steps and execution times, when you change the

stopping criterion and the absolute error bound?

Your project report also MUST

$[3$p$]$ contain at least one figure $($if your problem is $2-$dimensional, then your figure must include the all the steps starting from THREE random initial points with DIFFERENT COLORS and all the steps corresponding to the same iteration MUST be plotted with the SAME COLOR!$),$

$[3$p$]$ contain at least one table for benchmark,

$[4$p$]$ be at least two pages long and be written by IEEE conference proceeding template $[3]!$