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.)