Now, we present another type of vectorial subdifferential which extends to the vectorial case the Clarke subdifferential of real functions. To begin with, let us recall the definition of the circatangent cone.
Definition 7.1 Let C be a subset of a normed space Z and x belong to the closure of C.
The Clarke tangent cone or circatangent cone TC(x) is defined by
TC(x) ={v ∈Z | ∀hn↓0, ∀xn→C x, ∃vn →v, such that,∀n, xn+hnvn∈C}. Remark 7.1 Since Tepif(x, f(x)) is a convex, closed cone, then the directional deriva-tive denoted C↑f(x)(·) is a closed convex process. Thus, if X, Z are Banach spaces and dom C↑f(x) = X, then, fromTheorem2.2.6 [1], the multifunction u −−→−→ C↑f(x)(u) is pseudo-Lipschitz.
Remark 7.2 According to Proposition 6.2.3 [1], if f :X →Z is a Lipschitz map around a point x∈domf, then for a neighborhood U of x, the multifunction
(y, u)∈ U ×X −−→−→ C↑z∗◦f(x)(u) is upper semicontinuous and the multifunction
u −−→−→ C↑z∗◦f(y)(u) is pseudo-Lipschitz around x.
Obviously, if (u, v)∈Tepif(x), then (u, v+Z+)∈Tepif(x). As usual, we make the observation that if (x∗, z∗)∈Tepi◦ f(x), thenz∗ ∈ −Z+∗.
Definition 7.2 Let f : X → Z be a lower semicontinuous function and let z∗ ∈ Z+∗ \ {0}, x∈domf. We denote
∂zCl∗f(x) ={T ∈ L(X, Z)| (z∗◦T,−z∗)∈Tepi◦ f(x)} (37)
∂zCl∗ f(x) = {T ∈ L(X, Y Z)| (z∗◦T,−1) ∈Tepi◦ z∗◦f(x)} (38)
∂z∞∗ Clf(x) ={T ∈ L(X, Z)| (z∗◦T,0)∈Tepi◦ z∗◦f(x) (39) and for z∗ ∈Z∗ \ {0} we note
∂z∞∗Clf(x) ={T ∈ L(X, Z)| (z∗◦T,0) ∈Tepi◦ f(x). (40) The set given in (37) is called the Clarke vectorial subdifferentialof f at xand the set given in (38) is called theClarke scalar subdifferential off at x. The set given in (40) is called the singular vector Clarke subdifferential and the set given in (37) is called the singular scalar Clarke subdifferential.
We remark that if dom f =X, we can define (38) and (39) forz∗ ∈Z∗\ {0}.
Remark 7.3 Let z∗ ∈ Z∗ \ {0}. Denote by ∂Clz∗◦f(x) the usual Clarke subdifferential of the real function z∗◦f. Then for x∈domf
∂zCl∗ f(x) ={T ∈ L(X, Z)| z∗◦T ∈∂Clz∗◦f(x)} and thus, z∗◦∂zCl∗ f(x) =∂Clz∗◦f(x).
Proposition 7.1 Letf :X →Z be a lower semicontinuous function. For z∗ ∈Z+∗ \ {0} we have:
(u, v)∈Tepif(x) =⇒ (u, z∗(v))∈Tepiz∗◦f(x)(x).
Proof. Let (u, v) ∈ Tepif(x, f(x)). Thus for all hn ↓ 0 and (xn, yn) → (x, f(x)) we find (un, vn)→(u, v) such that
f(xn+hnun)≤yn+hnvn.
Let now hn↓0 and (xn, zn)→(x, z∗◦f(x)). For e∈Z such thatz∗(e) = 1, the sequence yn =zne+f(x)−(z∗ ◦f(x))e→ f(x).
Thus, there exists (un, vn)→ (u, v) such that
f(xn+hnun)≤yn+hnvn. This relation implies for z∗◦Z+∗ that
z∗◦f(xn+hnun)≤zn+hnz∗(vn) (41) and thus, there exists (un, z∗(vn))→(u, z∗(v)) such that (41) does hold, so the proposition
is proved. ✷
Corollary 7.1 If f : X → Z is a lower semicontinuous function, then for z∗ ∈ Z+∗ \ {0} and x∈domf we have that
∂zCl∗ f(x)⊆∂zCl∗f(x).
Proof. Use the previous proposition to derive that
z∗◦∂zCl∗ f(x)⊆z∗◦∂zCl∗f(x).
Thus, from the definition of the scalar Clarke subdifferential we obtain the expected inclusion.
✷
IfZis finite dimensional andf :X →Z is locally Lipschitz aroundx0, thenC↑z∗◦f(x0) = z∗◦C↑f(x0) for allz∗ ∈Z+∗ \ {0}and thus ∂zCl∗f(x0) =∂zCl∗ f(x0).
Proposition 7.2 Let f : X → Z be a lower semicontinuous function and z∗ ∈ Z+∗ \ {0}. Then, for x∈domf
∂z∗f(x)⊆∂zco∗f(x)⊆∂zCl∗f(x) and
∂z∗f(x)⊆∂zco∗f(x)⊆∂zCl∗ f(x).
If f is a convex, proper, lower semicontinuous function, then equality holds in these inclu-sions.
Proof. The following inclusions between tangent cones
Cepif(x)⊇Kepi f(x)⊇Tepi f(x), yields
z∗◦∂z∗f(x)⊆z∗ ◦∂zco∗f(x)⊆z∗◦∂zCl∗f(x).
Thus,
∂z∗f(x)⊆∂zco∗f(x) +Mz∗ ⊆∂zCl∗f(x) +Mz∗, where Mz∗ ={T ∈ L(X, Z)| z∗◦T = 0}.
Since ∂zco∗f(x) +Mz∗ =∂zco∗f(x) and ∂zCl∗f(x) +Mz∗ =∂zCl∗f(x), we derive that
∂z∗f(x)⊆∂zco∗f(x)⊆∂zCl∗f(x).
Similarly we obtain
∂z∗f(x)⊆∂zco∗f(x)⊆∂zCl∗ f(x).
If f is convex, then all concepts of tangent cones coincide and thus equality holds. ✷ It is also well-known (see [70]) that for a lower semicontinuous function g : X → IR, x∈domf,
sup
x∗∈∂Clg(x)
x∗(u) =g↑(x, u).
For vector mappings, we obviously have the following result.
Proposition 7.3 Let f : X → Z be a lower semicontinuous function and z∗ ∈ Z+∗, x ∈ domf. Then,
(z∗◦f)↑(x, u) = sup
T∈∂z∗Clf(x)
z∗◦T(u).
We observe that if the interior of the cone Z+ is nonempty, then for z∗ ∈ Z+∗ \ {0} we have
T ∈∂zCl∗f(x0) =⇒T u−v /∈IntZ+, ∀v ∈C↑f(x0)(u), u∈X.
Thus, if we denote
w∂Clf(x0) ={T ∈ L(X, Z)| T u>K v, ∀v ∈C↑f(x0)(u), u∈X},
we derive that
w∂Clf(x0)⊇
z∗∈Z+∗\{0}
∂zCl∗f(x0).
In fact, since the multifunctionx −−→−→ C↑f(x) is convex and since
w∂Clf(x0) ={T ∈ L(X, Z)| GrT ∩Gr C↑f(x0) =∅},
then for T ∈ w∂Clf(x0) we can find from the Hahn-Banach separation theorem some z∗ ∈ Z+∗ \ {0}, such that
z∗◦T u≤z∗(v), ∀(u, v)∈GrC↑f(x0).
This implies that T ∈∂zCl∗f(x0).
We can formulate now, the following proposition:
Proposition 7.4 Let f : X → Z be a lower semicontinuous function and suppose that the interior of the cone Z+ is nonempty. Then for x∈domf
w∂Clf(x0) =
z∗∈Z+∗\{0}
∂zCl∗f(x0).
Remark 7.4 For x0 ∈domf, we always have
∂≤f(x0)⊆
z∗∈Z∗+\{0}
∂zCl∗f(x0) and
w∂>f(x0)⊆w∂Clf(x0).
Now, as usually, we shall consider the Clarke coderivative given by:
C∗f(x)(z∗) ={x∗ ∈X∗ | (x∗,−z∗)∈Tepi◦ f(x)}. Thus, for z∗ = 0
z∗◦∂zCl∗f(x) =C∗f(x)(z∗).
Forz∗ = 0
C∗f(x)(0) =
z∗∈Z∗\{0}
z∗◦∂z∞∗Clf(x) =z∗◦∂z∞∗ Clf(x) for allz∗ ∈Z∗\ {0}.
Proposition 7.5 If X, Z are finite dimensional spaces, f :X → Z is a lower semicontin-uous function such that ∂zco∗f(y) = ∅ for y ∈ V ∈ V(x), x ∈ dom f and z∗ ∈ Z+∗ \ {0}, then
Limsup
y→x, f(y)→f(x)
∂zco∗f(y)⊂∂zCl∗f(x).
Proof. Under our assumption, Theorem 4.4.3 [1] shows that functions from X to Y with the weak topology, the pointwise convergence)
Proof. Since z∗ ◦f is a real Lipschitz functions around x, then ∂Clz∗◦f(x)= ∅ and thus,
∂zCl∗ f(x) is a nonempty set. Using the property of the polar cone Tepi◦ f(x) we derive that
∂zCl∗ f(x) is a convex, closed subset of Lp(X, Zw). The boundedness of ∂Clz∗ ◦f(x) implies that there existsmz∗ >0 such that for all T ∈∂zCl∗ f(x) we have z∗◦T ≤mz∗. ✷ The following result extends to the vector setting the theorem of Jofr´e- Correa-Thibault which expresses the convexity of a function in terms of the monotonicity of its Clarke subd-ifferential.
Proposition 7.7 Let X and Z be reflexive Banach spaces and let Z be ordered by a closed, convex pointed and proper coneZ+ which has a compact base. Letf :X →Z be a continuous mapping. The following assertions are equivalent : Combining the result of Correa and all ([13] Theorem 3.8) and Lemma 3.1, we obtain that
f is convex, establishing the proof. $
Corollary 7.1 By the same hypothesis like in Proposition 7.7 we obtain that f is convex if and only if ∂zCl∗ f(x) =∂≤f(x) +Mz∗ for all z∗ ∈Z+∗ \ {0}, for all x∈X.
If Z = IR, Proposition 7.7 says that if X is a reflexive space and f a continuous function, thenf is convex if and only if∂≤f(x) =∂Cf(x).We recover this result from [13] Proposition 2.3 and Theorem 3.8.
We recall now the definition of a monotone operator.
Definition 7.3 We say that the operator ∆ : D ⊂ X −−→−→ L(X, Z) is monotone if for all x, y∈D and T ∈∆(x), U ∈∆(y) we have
x−y, T −U ≥0.
We say that ∆ is a maximal monotone operator if it is maximal in the set of the monotone operatorsV :D⊂X −−→−→ L(X, Z).
As a supplementary remark linking the monotonicity of the subdifferential and the con-vexity for a vector-valued mapping, we have the following property.
Proposition 7.8 Let X and Z be reflexive Banach spaces and suppose that Z is ordered by a closed convex pointed and proper coneZ+. Iff:X→Z is a lower semicontinuous function (l.s.c.), then f is convex if and only if ∂C(z∗◦f)is a monotone operator for all z∗ belonging to Z+∗.
Proof. From Lemma 3.2, since f is l.s.c. we know that z∗◦f is l.s.c. for all z∗ ∈Z+∗. Then Theorem 3.8 in [13] ensures that the convexity of z∗◦f is equivalent with the monotony of
∂C(z∗◦f) for all z∗ ∈Z+∗. By virtue of Lemma 3.1,f is convex if and only ifz∗◦f is convex
and the proof is complete. $
Corollary 7.2 By the same hypothesis like in Proposition 7.8 we derive that f is convex iff
∂z∗f(x) = ∂zCl∗ f(x) for all x∈X, z∗ ∈Z+∗ \ {0}.
Proposition 7.9 Let f : X → Z be a proper lower semicontinuous function and z∗ ∈ Z+∗ \ {0}. If ∂zCl∗f is monotone, then z∗◦f is convex.
Conversely, if f is convex, then
z∗∈Z∗+\{0}∂zCl∗f is monotone.
Proof. If f is a proper, lower semicontinuous, convex function, then Tepif(x) =Kepif(x) = Sepif(x) and thus,
z∗∈Z+∗\{0}∂zCl∗ f =∂≤f which is obvious monotone.
Now, if∂zCl∗ f is monotone, thenz∗◦∂zCl∗ f is monotone and thus, ∂Clz∗◦f is monotone. From
Lemma 3.1 and [13] we obtain that z∗◦f is convex. ✷
We introduce now a vectorial version of the Zagrodny mean value Theorem for the non-smooth functions and its proof follows the Thibault’s one adapted to the vectorial case.
Theorem 7.1 Let two normed spaces (X,·) and (Z,·) separable, ordered by a pointed, closed, convex cone Z+. Let a lower semicontinuous function f : X → Z∪ {+∞}. Given two separate points in the domain of f, z∗ ∈Z+∗ \ {0} and ε ∈ Z+\ {0}, there exists a net (xn)n∈IN converging to c in [a, b) such that lim
n→+∞ z∗ ◦f(xn) = z∗ ◦ f(c), and there exists Tn ∈∂zCl∗ f(xn) such that for each n,
1. Tn, b−xn/b−xn ≥ −ε/n+ (f(b)−f(a))/b−a, 2. Tn, b−a/b−a ≥ −ε/n+ (f(b)−f(a))/b−a, 3. b−a z∗, f(c)−f(a) ≤ c−a z∗, f(b)−f(a).
Proof. We may suppose that f(b) =f(a) (else we consider the function g(x) =b−af(x) +b−x(f(b)−f(a))
which satisfies our condition). Since (Z,·) is a normed space, the quasi-interior Z+# ⊆ Z+∗ \ {0} is nonempty, and for eachz∗ ∈Z+∗ \ {0}, z∗◦f is a lower semicontinuous function (see Lemma 3.2). We can find a pointc∈[a, b) which is a minimum point for z∗◦f on [a, b]
and r >0 such that z∗◦f is lower bounded on U = [a, b] +rB (letγ be this lower bound).
We consider fU(x) =f(x) forx∈U and fU(x) = +∞, else. For each n, letrn ∈(0, r) such that for x∈[a, b] +rnB
z∗◦f(x)≥z∗◦f(c)− z∗, ε n2 and lettn≥n and u inZ+\ {0} such that z∗, u= 1 and
γ+tnz∗, urn ≥z∗◦f(c)−z∗, ε n2 . We obtain
z∗◦f(c)≤lim
x∈X z∗◦fU(x) +tnz∗, ud[a,b](x) + z∗, ε n2
(here d[a,b](·) is the distance function to [a, b] and B is the closed unit ball of X).
We can apply the Eckeland variational principle for the lower semicontinuous function Fn=z∗◦(fU +tnud[a,b]),
and we getxn ∈X such that
• i) c−xn ≤1/n,
• ii)Fn(xn)≤Fn(c) =z∗◦f(c),
• iii)∀x∈X, Fn(xn)≤Fn(x) + 1nz∗, ε x−xn.
From i), we may suppose that xn belongs to the interior of U. Now, iii) gives that 0∈∂Cl(Fn+ z∗, ε
n · −xn)(xn).
Using the Clarke subdifferential’s properties we derive:
0∈∂Clz∗◦f(xn) +∂Cl(z∗◦ ε
n · −xn)(xn) +∂Cltnz∗, ud[a,b](xn).
Since the distance function and the norm are convex and continuous, we get 0∈∂Clz∗◦f(xn) +∂≤(z∗◦ ε
n · −xn)(xn) +∂≤tnz∗, ud[a,b](xn).
Thus, there exist u∗n ∈ ∂Clz∗ ◦f(xn), vn∗ ∈ ∂≤d[a,b](xn) and b∗n ∈ B∗ (B∗ is the unit ball of the X∗) such that
−u∗n=z∗, tnuvn∗ + ε nb∗n.
We denote Tn =−tnuvn∗ −ε/nb∗n and using the definition of ∂zCl∗ f, we have that Tn ∈∂zCl∗ f.
If we choose yn ∈ [a, b] with xn−yn = d[a,b](xn) we have from i) that yn → c and moreover
vn∗, b−xn ≤d[a,b](b)−d[a,b](xn) =−d[a,b](xn)≤0, and sincev∗n ≤1,
vn∗, b−yn=vn∗, b−xn+vn∗, xn−yn ≤d[a,b](b)−d[a,b](xn) +v∗n xn−yn
≤ −d[a,b](xn) +d[a,b](xn) = 0
As yn →c =b, for n large enough we have yn ∈ [a, b) and the precedent inequality implies that
vn∗, b−a ≤0.
Since b∗n ∈B∗, we have
b∗n, b−xn ≤ b−xn, b∗n, b−a ≤ b−a. We derive for eachn,
Tn, b−xn ≥ −ε
n b−xn, Tn, b−a ≥ −ε
n b−a.
The third relation follows from iii). ✷
Remark 7.1 This result rest valid if we replace the lower semicontinuity of f with the condition that z∗◦f is lower semicontinuous for each z∗ ∈Z+∗ \ {0}.
Remark 7.2 If Z = IR, we refind the real Zagrodny mean theorem by passing to theliminf in the relations 1,2 and by taking the subsequences.
Using this theorem, we can given a characterization of the monotony of ∂zCl∗ f.
Corollary 7.3 Let two normed spaces (X,·)and (Z,·) separable, ordered by a pointed, closed, convex cone Z+ and let a lower semicontinuous function f :X →Z∪ {+∞}. Then, for z∗ ∈Z+∗ \ {0}, ∂zCl∗ f is monotone if and only if ∂zCl∗ f(x) =∂≤f(x), for all x∈X.
Proof. Obviously, if ∂zCl∗ f(x) =∂≤f(x), for all x∈X, then ∂zCl∗ f is monotone.
Now, if ∂zCl∗ f(x) =∅, then the equality follows using the fact that∂zCl∗ f(x)⊇∂≤f(x).
Let x ∈ X such that ∂zCl∗ f(x) = ∅. Then, x ∈ dom f and we can apply the vectorial version of the Zagrodny theorem for each d such that x+d ∈ dom f. Thus, there exists vk→d, tk→t∈(0,1] and Tk∈∂zCl∗ f(x+tkvk) such that
f(x+d)−f(x)
d ≥ Tk, vk vk − ε
k. Using the monotony of∂zCl∗ f, we get for T ∈∂zCl∗ f(x) and k that
f(x+d)−f(x)
d ≥ T, vk vk − ε
k. By taking the limit for k→+∞, we get
T, d ≤f(x+d)−f(x), ∀d∈domf −x
so, T ∈∂≤f(x) and the equality follows since we always have that ∂≤f(x)⊆∂zCl∗ f(x). ✷ Corollary 7.4 Let two normed spaces (X,·)and (Z,·) separable, ordered by a pointed, closed, convex cone Z+ with nonempty interior and let a lower semicontinuous function f : X → Z ∪ {+∞}. Then we have that ∂zCl∗ f is monotone for each z∗ ∈ Z+∗ \ {0} if and only if f is convex and ∂≤f(x) =w∂>f(x) for each x∈X.
Proof. From the precedent corollary we get that∂zCl∗ f is monotone for eachz∗ ∈Z+∗ \{0} if and only if for all x∈X
z∗∈Z+∗\{0}
∂zCl∗ f(x) =∂≤f(x).
Proposition 7.9 implies that z∗◦f is convex for each z∗ ∈Z+∗ \ {0}, thusf is convex. For a convex function we say that ∂zCl∗ f(x) =∂z∗f(x) and Proposition 4.5 gives that
w∂>f(x) =
z∗∈Z+∗\{0}
∂z∗f(x).
Thus, we derivew∂>f(x) =∂≤f(x).
Now, if f is convex, the equalityw∂>f(x) =∂≤f(x) implies that
z∗∈Z+∗\{0}
∂zCl∗ f(x) =∂≤f(x)
and the precedent corollary gives the conclusion. ✷
Corollary 7.5 If, in addition, we suppose in the precedent corollary that ∂≤f(x) = ∅ for each x ∈ dom f, then ∂zCl∗ f is monotone for each z∗ ∈ Z+∗ \ {0} if and only if f is convex and the order is total.
The proof follows from the precedent corollary and from Proposition 4.6.
The least result from this section is a simply consequence of the variational principle of Eckeland.
Proposition 7.10 Consider X, Z be Banach spaces and suppose that Z is ordered by a convex, closed, pointed, proper cone with nonempty interior. Let f : X → Z+ ∪ {+∞}
be a lower semicontinuous function, bounded from bellow, z∗ ∈ Z+∗ \ {0}, x ∈ dom f. If infD↑z∗ ◦f(x)(u) = −∞ for all u ∈ X, then for all ε ∈ IntZ+\ {0} there exists xε such that
z∗ ◦f(xε) +z∗(ε)xε−x0 ≤z∗◦f(x0) and
0∈∂zCl∗ f(xε) +εB∗. (Here B∗ denotes the closed unit ball).
Proof. Use Theorem 6.4.4 [1] to obtain for all ε ∈ IntZ+\ {0} the existence of some xε satisfying
z∗ ◦f(xε) +z∗(ε)xε−x0 ≤y∗◦f(x0) and
0∈z∗(∂zCl∗ f(xε) +εB∗).
The latter relation implies that there existx∗ ∈ B∗ and T ∈∂zCl∗ f(xε) such that z∗(T +εx∗) = 0.
This implies that −εx∗ ∈∂zCl∗ f(xε), and thus,
0∈∂zCl∗ f(xε) +εB∗.
✷