Consider the following nonlinear programming problem:

Maximize f(x)

x2 x

1 1, subject to x1  x2 2 and x1  0, x2  0.

(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.)