Consider the following linearly constrained programming problem:
Minimize f(x) = x31 + 4x22 + 16x3,
subject to
x1 + x2 + x3 = 5 and
x1 ≥ 1, x2 ≥ 1, x3 ≥ 1.
(a) Convert this problem to an equivalent nonlinear programming problem that fits the form given at the beginning of the chapter (second paragraph), with m = 2 and n = 3.
(b) Use the form obtained in part (a) to construct the KKT conditions for this problem.
(c) Use the KKT conditions to check whether (x1, x2, x3) = (2, 1, 2) is optimal.