Iterative Optimization in Inverse Problems

Geometric Programming and the MART 211

then we must have

g

1

(t

∗

)=1, (14.43)

and equality in the GAGM Inequality. Consequently,

3

2

40

t

∗

1

t

∗

2

t

∗

3

=3(40t

∗

2

t

∗

3

)=60, (14.44)

and

3

2

t

∗

1

t

∗

3

=

3

4

t

∗

1

t

∗

2

= K. (14.45)

Since g

1

(t

∗

)=1,wemusthaveK =

3

2

.Wesolvetheseequationsbytaking

logarithms, to obtain the solution

t

∗

1

=2,t

∗

2

=1, and t

∗

3

=

1

2

. (14.46)

The change of variables t

j

= e

x

j

converts the constrained (GP) prob-

lem into a constrained convex programming problem. The theory of the

constrained (GP) problem can then be obtained as a consequence of the

theory for the convex programming problem.

See [28] for a discussion of the use of constrained GP to find the Perron-

Frobenius eig ...