Consider the following convex programming problem. Maximize f( x ) = 10x 1 - 2 x 1 2 - x 1 3 + 8 x 2 -x 2 2 subject to x 1 + x 2 2 and x 1 0, x 2 0.

Consider the following convex programming problem.

Maximize f(x) = 10x1 - 2x12 - x1 3 + 8x2 -x22

subject to

x1 + x2 ≤ 2

and

x1 ≥ 0, x2 ≥ 0.

(a) Use the KKT conditions to demonstrate that (x1, x2) = (1, 1) is not an optimal solution.

(b) Use the KKT conditions to derive an optimal solution.