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

This preview shows pages 1–2. Sign up to view the full content.

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 (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online