Iterative Optimization in Inverse Problems

206 Iterative Optimization in Inverse Problems

so that, from Equation (14.22), we have

0=

n



i=1

a

ij

u

i

(t

∗

), (14.24)

for each j =1, ..., m. It follows that δ

∗

is feasible. Since we have equality

in the GAGM Inequality, we know

g(t

∗

)=v(δ

∗

). (14.25)

Therefore, δ

∗

solves (DGP). This completes the proof.

14.6 Solving the GP Problem

The theorem suggests how we might go about solving (GP). First, we

try to find a feasible δ

∗

that maximizes v(δ). This means we have to find

a positive solution to the system of m + 1 linear equations in n unknowns,

given by

n



i=1

δ

i

=1, (14.26)

and

n



i=1

a

ij

δ

i

=0, (14.27)

for j =1, ..., m, such that v(δ) is maximized. As we shall see, the multi- ...