Numerical Methods and Optimization

Unconstrained Optimization 337

If we premultiply both sides of this equation by (d

(i)

)

T

Q, we obtain

c

0

(d

(i)

)

T

Qd

(0)

+c

1

(d

(i)

)

T

Qd

(1)

+...+c

i

(d

(i)

)

T

Qd

(i)

+...+c

k

(d

(i)

)

T

Qd

(k)

=0.

But the directions d

(0)

,d

(1)

,...,d

(k)

are Q-conjugate, so (d

(i)

)

T

Qd

(j)

=0for

all j = i, and hence we have

c

i

(d

(i)

)

T

Qd

(i)

=0.

This means that c

i

=0,sinceQ is positive definite and d

(i)

= 0. Note that

the index i was chosen arbitrarily, hence c

i

=0foralli =1,...,k.

The linear independence of the conjugate directions implies that one can

choose at most nQ-conjugate directions in IR

n

.

13.6.1 Conjugate direction method for convex quadratic

problems

We consider a convex quadratic problem

min

x∈IR