Consider the following nonlinear programming problem: Maximize f(x) = X^2 + 1' subject to
X - 4^2 = 2 and
4^20 = 420
(a) Use the KKT conditions to demonstrate that (X1, X2) = (4,2) is not optimal.
(b) Derive a solution that does satisfy the KKT conditions.
(c) Show that this problem is not a convex programming problem.
(d) Despite the conclusion in part (c), use intuitive reasoning to show that the solution obtained in part (b) is, in fact, optimal. [The theoretical reason is that f(x) is pseudo-concave.]
(e) Use the fact that this problem is a linear fractional programming problem to transform it into an equivalent linear programming problem. Solve the latter problem and thereby identify the optimal solution for the original problem. (Hint: Use the equality constraint in the linear programming problem to substitute one of the variables out of the model, and then solve the model graphically.)