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.

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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) = 10×1 – 2×12 – x1 3 + 8×2 -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.
 
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