$($a$)$ Consider the convex program:

min,$f(x)$

$s.t.Axb$

$,x0$

Let $x^{(k)}$ be the $k$ th iterate in the Frank$-$Wolfe algorithm. Suppose the

optimal solution hat$(x{)}^{(k)}$ of $L{P}_k$ is $x^{(k)},$ prove that $x^{(k)}$ is an optimal solution

of the convex program.

$($b$)$ Consider the following convex program:

min,$f(x)$:$=(x_1-2{)}^2+3x_2^2$

$s.t.x_1+3x_{2}6$

$,x_1+x_{2}4$

$,x_1,x_{2}0.$

$($i$)$ Starting from the initial point $x^{(0)}=\left[\begin{array}{c}2\\ 2\end{array}\right],$ use the Frank$-$Wolfe

algorithm to find the iterates $x^{(1)},x^{(2)},x^{(3)}.$

$($ii$)$ Using Part $($a$),$ determine an optimal solution of the convex program.