# Question

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.

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.

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