ISyE 6663 Optimization III
Spring 2003
Assignment 5
Issued: March 13, 2003
Due: March 20, 2003
Problem 1
Nocedal and Wright, Problem 5.1
Problem 2
Nocedal and Wright, Problem 5.4
Problem 3
Nocedal and Wright, Problem 5.11
Problem 4
Consider a symmetric positive definite matrix Q∈Rn×n, and the associated norm xQ:= xTQx.
Consider Q-conjugate directions d0,d
1,...,d
n−1∈Rngenerated from linearly independent vectors
p0,p
1,...,p
n−1∈Rn. Show that, for each k=1,...,n−1, dk=pk−ˆpk, where ˆpkis the projection
of pkonto the subspace spanned by p0,...,p
k−1(or the subspace spanned by d0,...,d
k−1)with
respect to the ·
Q-norm, that is,
ˆpk=argmin{pk−pQ:p∈[p0,...,p
k−1]}
That is, dkis the part of pkthat remains after we subtract the projection of pkonto the subspace
spanned by p0,...,p
k−1.