25.6. GENERALIZED GRADIENTS 865

25.6 Generalized GradientsThis is an interesting theorem, but one might wonder if there are easy to verify examples ofsuch possibly set valued mappings. In what follows consider only real spaces because theessential ideas are included in this case which is also the case of most use in applications.Of course, you might with some justification, make the claim that the following is not reallyvery easy to verify any more than the original definition.

Definition 25.6.1 Let V be a real reflexive Banach space and let f : V → R be a locallyLipschitz function, meaning that f is Lipschitz near every point of V although f need notbe Lipschitz on all of V. Under these conditions,

f 0 (x,y)≡ lim supµ→0+ h→0

f (x+h+µy)− f (x+h)µ

(25.6.31)

and ∂ f (x)⊆ X ′ is defined by

∂ f (x)≡{

x∗ ∈ X ′ : x∗ (y)≤ f 0 (x,y) for all y ∈ X}. (25.6.32)

The set just described is called the generalized gradient. In 25.6.31 we mean the followingby the right hand side.

lim(r,δ )→(0,0)

sup{

f (x+h+µy)− f (x+h)µ

: µ ∈ (0,r) ,h ∈ B(0,δ )}

I will show, following [99], that these generalized gradients of locally Lipschitz func-tions are sometimes pseudomonotone. First here is a lemma.

Lemma 25.6.2 Let f be as described in the above definition. Then ∂ f (x) is a closed,bounded, convex, and non empty subset of V ′. Furthermore, for x∗ ∈ ∂ f (x) ,

||x∗|| ≤ Lipx ( f ) . (25.6.33)

Proof: It is left as an exercise to verify the assertions that ∂ f (x) is closed, and convex.It follows directly from the definition. To verify this set is bounded, let Lipx ( f ) denote aLipschitz constant valid near x ∈V and let x∗ ∈ ∂ f (x) . Then choosing y with ||y||= 1 andx∗ (y)≥ 1

2 ||x∗|| ,

12||x∗||= x∗ (y)≤ f 0 (x,y) . (25.6.34)

Also, for small µ and h,∣∣∣∣ f (x+h+µy)− f (x+h)µ

∣∣∣∣≤ Lipx ( f ) ||y||= Lipx ( f ) .

Therefore, f 0 (x,y)≤ Lipx ( f ) and so 25.6.34 shows ||x∗|| ≤ 2Lipx ( f ) .The interesting part of this Lemma is that ∂ f (x) ̸= /0. To verify this first note that the

definition of f 0 implies that y→ f 0 (x,y) is a gauge function. Now fix y ∈V and define onRy a linear map x∗0 by

x∗0 (αy)≡ α f 0 (x,y) .