$(1)(8$ points$)$ Write a QP subproblem at the starting point in terms of the search direction $d.$ Assume the

Hessian for the Lagrange function is the identity matrix at the initial iteration.

$(2)(14$ points$)$ Write the KKT condition for the subproblem and find the optimum solution for $d$ and

Lagrange multipliers $({}_i).$

$(3)(8$ points$)$ Find the optimal search distance $$ and $x_j.$ Use the penalty term $(R)$ as $R={}_i(|{}_i|).$

$(4)(5$ points$)$ Solve this optimization problem using MATLAB fimincon.

$(5)(5$ points$)$ Provide a plot $($MATLAB$)$ including the contour of $f,$ the boundaries of $g_i,$ and indicate $x_0,$

$x_1,$ and $x^{**}($optimal point$)$ in one plot.