Problem $2.$

i$).$ Solve the following Portfolio Management problem:

minimize $w^T\left[\begin{array}{ccccccccccccc}0\mathrm{.}7& 0\mathrm{.}2& 0\mathrm{.}1& 0\mathrm{.}1& 0\mathrm{.}5& 0\mathrm{.}2& 0\mathrm{.}3& 0\mathrm{.}2& 0\mathrm{.}4& 0\mathrm{.}01& 0\mathrm{.}3& 0\mathrm{.}01& 0\mathrm{.}4\end{array}\right]$

$0\mathrm{.}1$

$0\mathrm{.}1$

$0\mathrm{.}1$

$0\mathrm{.}1$

$0\mathrm{.}1$

$0\mathrm{.}2w$

Subject $to$: $w^T\left[\begin{array}{c}5\end{array}\right]$

$4$

$1$

$2=6,w^T\left[\begin{array}{c}1\end{array}\right]$

$1$

$1$

$1=1,w=\left[\begin{array}{c}{w}_1\end{array}\right]$

$w_2$

$w_3$

$w_4$

ii$).$ If in the first constraint, instead of $6$ we have $6+,$ where $$ is very

small in magnitude, estimate without resolving the problem, how much

the optimum value of the objective function changes. Use this estimate

for $=0\mathrm{.}1,0\mathrm{.}2,$ and $1.$ Calculate the exact optimum for $=1.$ Comment.