Week _13 solution

# Week _13 solution - Solutions to the assignment due on...

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6. Use Solver to maximize f(x, y, z) = x y z , subject to λ : 4 x y + 3 x z + 2 y z 72 , x 0 , y 0 , z 0 . Then write the KKT conditions for the same optimization problem, and solve them analytically. Do you get the same solution? Since the decision variables are nonnegative in this maximization problem, the cross-over table shows that the complementary constraints are these “≥” inequalities: x : λ (4 y + 3 z) y z , y : λ (4 x + 2 z) x z , z : λ (3 x + 2 y) x z . The optimal value is positive, we know that x , y and z are positive, so complementary slackness shows that the above three inequalities hold the equations: x : z 3 y 4 z y + = λ , y : z 2 x 4 z x + = λ , z : y 2 x 3 y x + = λ . Since x, y and z are positive, these equations guarantee that λ is positive, so complementary slackness shows that λ : 4 x y + 3 x z + 2 y z = 72 , That’s four equations in four unknowns. The right-hand sides of the first and second

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## This note was uploaded on 08/04/2011 for the course MATH 235 taught by Professor Ericdenardo during the Fall '10 term at Yale.

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Week _13 solution - Solutions to the assignment due on...

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