Iterative Optimization in Inverse Problems

62 Iterative Optimization in Inverse Problems

Proof: The first assertion follows from Equation (4.71) and continuity. We

have

h(x

k

)+

1

2

T (x

k

) − b

k−1



2

2

≤ h(ˆx)+

1

2

T (ˆx) −b

k−1



2

2

,

so that, by taking limits, we have

h(x

∗

)+

1

2

b

∗



2

2

≤ h(ˆx)+

1

2

b

∗



2

2

.

This theorem is similar to Theorem 2.2 of [132]; the latter does not

require that h be differentiable.

In the Goldstein-Osher algorithm we have

b

k

= b

k−1

− T (x

k

). (4.72)

Now we can strengthen Theorem 4.5.

Theorem 4.6 Let h be differentiable. Let b

k

be defined as in Equation

(4.72). If the sequence {x

k

} converges to some x

∗

,thenT (x

∗

)=0.Conse-

quently, x

∗

minimizes h(x) over x in S.Ifh is not differentiable, but the

sequenc ...