Iterative Optimization in Inverse Problems

Barrier-Function and Penalty-Function Methods 45

{x

k

} is bounded, so that there is a convergent subsequence; let {x

k

n

}→x

∗

.

It follows that p(x

∗

) = 0, so that x

∗

is in C.Then

f(x

∗

)=f(x

∗

)+p(x

∗

) = lim

n→+∞

(f(x

k

n

)+p(x

k

n

)) ≤ lim

n→+∞

T

k

n

(x

k

n

)=γ ≤ d.

But x

∗

∈ C,sof(x

∗

) ≥ d. Therefore, f(x

∗

)=d.

It may seem odd that we are trying to minimize f (x)overthesetC

using a sequence {x

k

} with {f (x

k

)} increasing, but remember that these

x

k

are not in C.

Definition 3.1 Let X be a complete metric space. A real-valued function

p(x) on X has compact level sets if, for all real γ, the level set {x|p(x) ≤ γ}

is compact.

Theorem 3.2 Let X be a complete metric space, f(x) be a continuou ...