Lecture 18 Subgradients

Lecture 18
convergent subsequence, say {d k } K ′ with K ′ ⊆ K. Let d be the limit of
{d k } K ′.
Since s k ∈ ∂f(x k ), we have for each k,
f(x k + d k ) ≥ f(x k ) + s T k d k = f(x k ) + ‖s k ‖.
By letting k → ∞ with k ∈ K ′ , we see that
lim sup[f(x k + d k ) − f(x k )] ≥ lim sup ‖s k ‖.
k→∞
k∈K ′
By continuity of f,
k→∞
k∈K ′
lim sup[f(x k + d k ) − f(x k )] = f(x + d) − f(x),
k→∞
k∈K ′
hence finite, implying that ‖s k ‖ is bounded. This is a contradition ({s k }
was assumed to be unbounded).
Convex Optimization 9