Iterative Optimization in Inverse Problems

70 Iterative Optimization in Inverse Problems

We want to minimize the function f

2

(x)overx in C, or, equivalently, to

minimize the function f(x)=ι

C

(x)+f

2

(x). The projected gradient descent

algorithm has the iterative step

x

k

= P

C



x

k−1

− γA

T

(I − P

Q

)Ax

k−1



; (5.18)

this iterative method was called the CQ-algorithm in [58, 59]. The sequence

{x

k

} converges to a solution whenever f

2

has a minimum on the set C,for

0 <γ≤ 1/L.

In [85, 81] the CQ algorithm was extended to a multiple-sets algorithm

and applied to the design of protocols for intensity-modulated radiation

therapy.

5.4.3 The Projected Landweber Algorithm

The problem is to minimize the function

f

2

(x)=

1

2

