Optimization In Operations Research

$3-14$ Consider the mathematical program

max $4$z$\_1+7$z$\_2$

s$.$t$.2$z$\_1+$ z$\_29$

$0$ z$\_14$

$0$ z$\_23$

$$

$($a$)$ Show that directions $$z$^1=(2,0)$ and $$z$^2=(-2,4)$ are improving directions for this model at every z$.($has been answered below$)$

Gradient f $=(4,7)$

$(4,7)$ dot $(2,0)=4*2+7*0=8>0$

$(4,7)$ dot $(-2,4)=4*-2+7*4=20>0$

$$

$($b$)$ Beginning at z$^2=(0,0),$ execute Improving Search Algorithm $3$A on the model. Limit your search to the two directions of part $($a$),$ and continue until neither is both improving and feasible. $$

$$

$($c$)$ Show in a two$-$dimensional plot the feasible space and objective function contours of the model. Then plot the path of your search in part $($b$).$