1380 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

The subgradient is an attempt to generalize the derivative. For example, a functionmay have a subgradient but fail to be differentiable at some point. A good example isf (x) = |x|. At x = 0, this function fails to have a derivative but it does have a subgradient.In fact, δ f (0) = [−1,1].

To begin with consider the question of existence of the subgradient of a convex func-tion. There is a very simple criterion for existence. It is essentially that the subgradientis nonempty at every point of the interior of the domain of φ . First recall Lemma 18.2.15which says the interior of a convex set is convex and if nonempty, then every point of theconvex set can be obtained as the limit of a sequence of points of the interior.

Theorem 42.3.2 Let φ : X → (−∞,∞] be convex and suppose for some u ∈ dom(φ), φ iscontinuous. Then δφ (x) ̸= /0 for all x ∈ int(dom(φ)). Thus

dom(δφ)⊇ int(dom(φ)).

Proof: Let x0 ∈ int(dom(φ)) and let

A≡ {(x0,φ (x0))} ,B≡ epi(φ)∩X×R.

Then A and B are both nonempty and convex. Recall epi(φ) can contain a point like (x,∞).Since φ is continuous at u ∈ dom(φ),

(u,φ (u)+1) ∈ int(epiφ ∩X×R) .

Thus int(B) ̸= /0 and also int(B)∩A = /0. By Lemma 18.2.15 int(B) is convex and so byTheorem 18.2.14 there exists x∗ ∈ X ′ and β ∈ R such that

(x∗,β ) ̸= (0,0) (42.3.18)

and for all (x,a) ∈ intB,x∗ (x)+βa > x∗ (x0)+βφ (x0). (42.3.19)

From Lemma 18.2.15, whenever x ∈ dom(φ),

x∗ (x)+βφ (x)≥ x∗ (x0)+βφ (x0).

If β = 0, this would mean x∗ (x− x0) ≥ 0 for all x ∈ dom(φ). Since x0 ∈ int(dom(φ)),this implies x∗ = 0, contradicting 42.3.18. If β < 0, apply 42.3.19 to the case when a =φ (x0)+1 and x = x0 to obtain a contradiction. It follows β > 0 and so

φ (x)−φ (x0)≥−x∗

β(x− x0)

which says −x∗/β ∈ δφ (x0). This proves the theorem.

Definition 42.3.3 Let φ : X → (−∞,∞] be some function, not necessarily convex but satis-fying φ (y)< ∞ for some y ∈ X. Define φ

∗ : X ′→ (−∞,∞] by

φ∗ (x∗)≡ sup{x∗ (y)−φ (y) : y ∈ X}.