$($MATLAB experiment$)$ Consider the following nonlinear optimization problem:

$mi{n}_{x,\text{y}\text{i}\text{n}\text{R}}f(x,y)=x^2-5xy+y^4-25x-8y.$

$($a$)$ Implement Gradient Descent with Armijo's rule for line search, and use it to find a relative

minimum point. Explain the parameters of the algorithm you used. Plot $($i$)||gradf(x_k)||$ versus $k$

$($use the $log$ scale for the $y-$axis so that you will be able to recognize the small difference such

as the one between ${10}^{-4}$ and ${10}^{-6}),$ and $($ii$)f(x_k)$ versus $k$ and provide interpretation. Check

the Hessian at the algorithm output to verify whether the algorithm output is indeed a relative

minimum point.

$($b$)$ Implement Gradient Descent with Goldstein rule for line search, and use it to find a relative

minimum point. Explain the parameters of the algorithm you used. Plot $($i$)||gradf(x_k)||$ versus $k,$

and $($ii$)f(x_k)$ versus $k$ and provide interpretation. Check the Hessian at the algorithm output to

verify whether the algorithm output is indeed a relative minimum point.