Consider the following nonlinear programming problem.Maximize Z = ,subject to 2x₁ + x₂ + x₃ ≤ 4 and x₁ ≥ 0,x₂ ≥ 0,x₃≥ 0.
(a)Use the KKT conditions to determine whether = (1,1,1)can be optimal.
(b)If a specific solution satisfies the KKT conditions for this problem,can you draw the definite conclusion that this solution is optimal? Why?

Question

Consider the following nonlinear programming problem.Maximize Z =

,subject to 2x₁ + x₂ + x₃ ≤ 4 and x₁ ≥ 0,x₂ ≥ 0,x₃≥ 0.
(a)Use the KKT conditions to determine whether

= (1,1,1)can be optimal.
(b)If a specific solution satisfies the KKT conditions for this problem,can you draw the definite conclusion that this solution is optimal? Why?

Quizplus · Accepted Answer

The Answer is The KKT conditions are 1 a) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 2 a) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 1 b) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 2 b) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 1 c) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 2 c) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 3) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 4) $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ 5)x₁

\ge

0,x₂

\ge

0,x₃

\ge

0,6)u

\ge

0.(a)Consider (x₁,x₂,x₃)= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x₁,x₂,x₃)= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 $The KKT conditions are 1 a) 2 a) 1 b) 2 b) 1 c) 2 c) 3) 4) 5)x1 \ge 0,x2 \ge 0,x3 \ge 0,6)u \ge 0.(a)Consider (x1 ,x2,x3 )= (1,1,1).From 2a),we have u = 2.Note that all the KKT conditions now are satisfied,so (x1 ,x2,x3 )= (1,1,1)can be optimal.(b)No.The objective function is not concave (since 2 is convex instead),so the KKT conditions are necessary but not sufficient for optimality.$ is convex instead),so the KKT conditions are necessary but not sufficient for optimality.

Consider the Following Nonlinear Programming Problem