Numerical Methods and Optimization

386 Numerical Methods and Optimization: An Introduction

where Q is a symmetric positive definite matrix, A is a m × n matrix,

and b ≥ 0.

14.15. Show that the dual function d(μ) defined in Eq. (14.12) at page 369 is

concave.

14.16. Apply two iterations of the affine scaling method to the following prob-

lem:

minimize 2x

1

+ x

2

+3x

3

subject to x

1

+2x

2

− x

3

=1

x

1

+ x

2

+ x

3

=1

x

1

,x

2

,x

3

≥ 0.

Use x

(0)

=[1/4, 1/2, 1/4]

T

as the initial point. At each iteration, use the

step-length coefficient α =0.9.