Please use PYTHON and MATLAB. Write a numerical code to minimize the Rosenbrock function defined below using a slew of gradient based optimization algorithms. The function is a non$-$convex function and has a global minimum at $(1,1).$ The minimum is at the bottom of a narrow parabolic valley that is curved on the x$-$y plane. The problem statement is as follows

minimize f$($x$,$ y$)=(1$ x$)^2+100($y $$ x$^2)^2$

with respect to x$,$ y in Rn

subject to $$c$1($x$,$ y$)=1$ x $$ y $=0$

$($or$)$c$2($x$,$ y$)=1$ x$^2$ y$^2=0$

$1.$ Solve the above optimization problem with only constraint $$c$1($x$,$ y$)$ by$-$hand and find x$.$ Show all the necessary steps and demonstrate that the first$-$order necessary conditions are satisfied.

$2.$ Write an SQP algorithm to find the minimum of the function subject to two single

equality constraints. Use a simple backtracking line search method. Employ a Newton$$s method. Use only one constraint at a time.

$3.$ Repeat the above with a Quadratic Penalty approach to satisfy the constraints.

$4.$ Provide the following in a written report for each of the constraints and optimization approaches $($SQP and Quadratic Penalty$)$:

$($a$)$ Convergence of the gradient $($y$-$axis: log of the gradient, x$-$axis: iteration$)$ and a comparison of the convergence.

$($b$)$ Contour plot of the path of the optimization algorithm.

$($c$)$ Discuss and compare the plots, as well as discuss the choice of parameters used

in your results and their effect on the optimization.