hw3 - erties that k A x k k A kk x k and k AB k k A kk B k...

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

View Full Document Right Arrow Icon
22M:174 Optimization Techniques Homework 3 — due Friday Feb 24 1. Implement the standard Newton method for solving f ( x ) = 0 without line search. Test it on the function f ( x , y ) = x 4 - x 2 y + 2 y 2 - 2 y with the starting points ( x , y ) = ( 0 . 01 , 0 . 1 ) , ( 1 , 1 2 ) , and ( - 10 , 5 ) . 2. Implement the standard Newton method with the modification that if p T k f ( x k ) 0 for p k = - 2 f ( x k ) - 1 f ( x k ) , then we set p k ← - f ( x k ) ; that is, we use steepest descent as a back-up method if p k is not a descent direction. Use a Wolfe-condition based line search (see ICON for Matlab code). Apply this method to f ( x , y ) = x 4 - x 2 y + 2 y 2 - 2 y with starting points ( x , y ) = ( 0 . 01 , 0 . 1 ) , ( 1 , 1 2 ) , and ( - 10 , 5 ) . 3. Matrix norms . If we have a norm for vectors k x k , we can use it to define a matrix norm: the norm of a matrix is the amount the matrix can amplify the size of the vector: k A k = max x 6 = 0 k A x k k x k . Different vector norms will give different matrix norms. Show that matrix norms have the usual properties: (a) k A k ≥ 0 and k A k = 0 implies that A = 0; (b) k α A k = | α |k A k for any scalar α ; (c) k A + B k ≤ k A k + k B k . Show, in addition, that the matrix norm also has the multiplication prop-
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: erties that k A x k k A kk x k and k AB k k A kk B k . If the vector norm is k x k : = max i | x i | , show that the corresponding matrix norm is k A k = max i j a ij . 4. The condition number of an invertible matrix A is dened as ( A ) = k A k A-1 . Show that for any invertible matrix A , ( A ) 1. [ Hint: Apply the multipli-cation property to k I k = AA-1 , and check that k I k = 1.] 1 5. Suppose g : R n R n has a Lipschitz continuous Jacobian matrix ( k g ( x )- g ( y ) k L k x-y k ). Show that g ( y ) = g ( x )+ g ( x )( y-x )+ Z 1 [ g ( x + s ( y-x ))- g ( x )]( y-x ) ds . Use this to show that k g ( y )-[ g ( x )+ g ( x )( y-x )] k 1 2 L k y-x k 2 . 2...
View Full Document

Page1 / 2

hw3 - erties that k A x k k A kk x k and k AB k k A kk B k...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online