Iterative Optimization in Inverse Problems

Eigenvalue Bounds 133

where, for any real vector a,wedenoteby(a)

+

the nonnegative vector

whose entries are those of a, for those that are nonnegative, and are zero

otherwise. The projected Landweber algorithm converges to a vector that

minimizes Ax − b

2

over all nonnegative vectors x, for the same values

of γ.

The projected Landweber algorithm is actually more general. For any

closed, nonempty convex set C in C

J

, define the iterative sequence

x

k+1

= P

C

(x

k

+ γA

†

(b −Ax

k

)). (9.35)

This sequence converges to a minimizer of the function Ax − b

2

over all

x in C, whenever such minimizers exist.

Both the Landweber and projected Landweber algorithms are special