08sfin - 1 1. Find all the local minimizers of the function...

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1 1. Find all the local minimizers of the function f ( x 1 , x 2 ) = x 2 1 + 1 3 x 3 2 x 1 x 2 and demonstrate that they satisfy the su±cient conditions presented in the class. 2. Using Newton’s method to ²nd the stationary points of f ( x 1 , x 2 ) = x 3 1 x 2 + 2 x 1 x 2 2 we start with ¯ x 0 = (1 , 0) T . Calculate ¯ x 1 in fraction form. 3. Prove complementary slackness in linear programming: Let x * be an optimal solution to the linear programming problem: Minimize z = c T x subject to Ax = b, x 0 and let y * be an optimal solution to the dual problem: Maximize w = b T y subject to A T y c , then x T * ( c A T y * ) = 0. 4. Given the problem Minimize f ( x 1 , x 2 ) = x 1 subject to ( x 1 + 1) 2 + x 2 2 1 x 2 1 + x 2 2 2 (a) Write the Lagrangian function L ( x 1 , x 2 , λ 1 , λ 2 ) of this problem as a function of four variables. (b) Show that ( 1 , 1 , 1 / 2 , 1 / 2) is a stationary point of L ( x 1 , x 2 , λ 1 , λ 2 ). (c) Show that (
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08sfin - 1 1. Find all the local minimizers of the function...

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