Problem $1$: The function

Comsider the N$-$D comes function

$f(x_1,$dots,$x_N)=2\mathrm{.}4{\displaystyle \underset{i=1}{\overset{N}{}}}{\displaystyle \underset{j=1}{\overset{N}{}}}\frac{x_i{x}_j}{i+j-1}-{\displaystyle \underset{i=1}{\overset{N}{}}}i{x}_i=1\mathrm{.}47$

and plot the graph of the function for $N=2.$

Problem $2$: Optimization problem

Express the function $(1)$ in the form $f(x)=\frac{1}{2}{x}^{7}Sx-b^{T}x+c$ for $N=8$ where $S=S^T.$

Problem $3$: Properties of $S$

For $N=8,$ show that $S$ is positive definine by checking its eigemalues.

Problem $4$: Implementation of the Gradient Descent

Find the global minimum of the function in $(1)$ for $N-8$ using the Gradiem Descent algorithm.

Take the initial poims as $x^{(9)}-{\left[\begin{array}{cccc}1\mathrm{.}5& 1\mathrm{.}5& \text{c}\text{d}\text{o}\text{t}\text{s}& 1\mathrm{.}5\end{array}\right]}^7$ and apply the following learning rate selection

approaches:

Inexact line search with $s_0=0\mathrm{.}1,0\mathrm{.}01$ and $0\mathrm{.}001,$ respectively.

Backtracking with $s_0=0\mathrm{.}5.$

Momentum with the opeinsal parameter.

Nesterov method with the optimal paramesers.

ADAM method with the default parameters.

Problem $5$: The Stochastic Gradient Descent

Find the global minimam of the function in $(1)$ for $N-8$ using the Stocustic Gradient Descent

algorithm with mini$-$batch size is $2.$

Problem $6$: Compare and justify your results

Plot a $2-$D graph including the values of the function in $(1)$ vs the iteration number for each

approach and method in Problem $4$ and Problem $5.$ Discuss your findings.