Iterative Optimization in Inverse Problems

52 Iterative Optimization in Inverse Problems

where p

k−1

is a member of ∂h(x

k−1

). Therefore,

0 ∈ ∂f(x

k

)+∂h(x

k

) − p

k−1

. (4.20)

We assume that this equation can be solved and that there are u

k

∈ ∂f(x

k

)

and v

k

∈ ∂h(x

k

)sothat

v

k

= p

k−1

− u

k

. (4.21)

We then define p

k

= v

k

,sothat

G

k

(x) − G

k

(x

k

)=

D

f

(x, x

k

; u

k

)+D

h

(x, x

k

; p

k

) ≥ D

h

(x, x

k

; p

k

)=g

k+1

(x). (4.22)

Therefore, the SUMMA condition holds and the sequence {f(x

k

)} con-

verges to f(ˆx).

4.7 The IPA

The IPA is a modification of the PMA designed to overcome some of

the computational obstacles encountered in the PMA [55, 63]. At the kth

step of the PMA we must solve the equation

∇f(x

k

)+∇h(x

k

)=∇h(x

k−1

) (4.23)

for x

k

, where, ...