23.6 Generalized Gradients
This is an interesting theorem, but one might wonder if there are easy to verify examples
of such possibly set valued mappings. In what follows consider only real spaces because
the essential 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 really very easy to verify any more than the original
Definition 23.6.1 Let V be a real reflexive Banach space and let f : V → ℝ be a locally
Lipschitz function, meaning that f is Lipschitz near every point of V although f need not
be Lipschitz on all of V. Under these conditions,
⊆ X′ is defined by
The set just described is called the generalized gradient. In 23.6.32 we mean the following
by the right hand side.
I will show, following [?], that these generalized gradients of locally Lipschitz
functions are sometimes pseudomonotone. First here is a lemma.
Lemma 23.6.2 Let f be as described in the above definition. Then ∂f
is a closed, bounded, convex, and non empty subset of V ′. Furthermore, for
Proof: It is left as an exercise to verify the assertions that ∂f
is closed, and
convex. It follows directly from the definition. To verify this set is bounded, let
denote a Lipschitz constant valid near
x ∈ V
and let x∗∈ ∂f
Then choosing y
= 1 and
Also, for small μ and h,
The interesting part of this Lemma is that ∂f
To verify this first note that the
definition of f0
implies that y → f0
is a gauge function. Now fix
y ∈ V
a linear map x0∗
Then if α ≥ 0,
If α < 0,
By the Hahn Banach theorem there is an
extension of x0∗
to all of V, x∗
for all y. It remains to verify x∗ is continuous. This follows easily from
which verifies 23.6.34 and proves the lemma.
This lemma has verified the first condition needed in the definition of pseudomonotone.
The next lemma verifies that these generalized subgradients satisfy the second of the
conditions needed in the definition. In fact somewhat more than is needed in the
definition is shown.
Lemma 23.6.3 Let U be weakly open in V ′ and suppose ∂f
⊆ U. Then
⊆ U whenever z is close enough to x.
Proof: Suppose to the contrary there exists zn → x but zn∗∈ ∂f
first lemma, we may assert that
large enough. Therefore,
there is a subsequence, still denoted by n
such that zn∗
converges weakly to
Proof of the claim: There exists δ > 0 such that if μ,
be so large that
choosing h ≡ h′−
it follows the above inequality holds because
is sufficiently large,
Consequently, for all n large enough,
which proves the claim.
Now with the claim,
so z∗∈ ∂f
contradicting the assumption that
This proves the lemma.
It is necessary to assume more on f0 in order to obtain the third axiom defining
pseudomonotone. The following theorem describes the situation.
Theorem 23.6.4 Let f : V → V ′ be locally Lipschitz and suppose it satisfies the
condition that whenever
it follows that
for all z ∈ V. Then ∂f is pseudomonotone.
Proof: 23.4.16 and 23.4.17 both are satisfied thanks to Lemmas 23.6.1 and 23.6.2. It
remains to verify 23.4.18. To do so, I will adopt the convention that x∗∈ ∂f
This implies liminf n→∞xn∗
which implies, by the above assumption that for all z,
In particular, this holds for z = x and this implies limsupxn∗
0 which along
Now let z be arbitrary. There exists a subsequence, nk, depending on z such
Now from Lemma 23.6.2 and its proof, the
are all bounded by
is large enough. Therefore, there is a further subsequence, still denoted by nk
We need to verify that x∗
To do so, let y
be arbitrary. Then from the
From 23.6.38, we can take the limsup of both sides and obtain, using 23.6.37
Since y is arbitrary, this shows x∗
and proves the theorem.