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# hw3 - erties that k A x k ≤ k A kk x k and k AB k ≤ k A...

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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-

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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 deﬁned as κ ( A ) = k A k ² ² A-1 ² ² . Show that for any invertible matrix A , κ ( A ) ≥ 1. [ Hint: Apply the multipli-cation property to k I k = ² ² AA-1 ² ² , 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...
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