Iterative Optimization in Inverse Problems

108 Iterative Optimization in Inverse Problems

We are used to thinking of functions that are not differentiable as lacking

something. From the point of view of subgradients, not being differentiable

means having too many of something.

The gradient of the function f (x) in Equation (8.1) is

∇f(x)=x −

1

I

I



i=1

P

C

i

x. (8.4)

Therefore, a gradient descent approach to minimizing f(x) has the iterative

step

x

k+1

= x

k

− γ

k



x

k

−

1

I

I



i=1

P

C

i

x

k



=

(1 − γ

k

)x

k

+ γ

k



1

I

I



i=1

P

C

i

x

k



. (8.5)

This is sometimes called the relaxed averaged projections algorithm.Aswe

shall see shortly, the choice of γ

k

= 1 is sufficient for convergence.

8.2 Using Orthogonal Projections

When the convex sets