$1.$ Solve the following asset allocation problem manually using the Linear Programming method

$2.$ Convert this problem into an unconstrained optimization problem using the log$-$barrier penalty method. Solve it using both the Conjugate Gradient Method and Quasi$-$Newton's Method.

Determine the optimal asset allocation to maximize return while controlling risk $(<=3\%)$ by investing in three stocks $($X$,$ Y$,$ Z$).$ It's important to note that full investment is not necessary. Below are the returns and risks associated with each stock

Stock X: Return $=10\%,$ Risk $=15\%$

Stock Y: Return $=8\%,$ Risk $=4\%$

Stock Z: Return $=3\%,$ Risk $=2\%$