For a lower semicontinuous function f :X →Z we denote for (x0, f(x0))∈epif Cepif(x0) = cl cone(epi f−(x0,f(x0))),
where cl cone K means the closure of the cone generated byK.
Cepif(x0) is a closed cone which is convex if f is convex. We denote Cepi◦ f(x0) the polar cone of Cepif(x0), that is:
Cepi◦ f(x0) ={(x∗, z∗)∈X∗ ×Z∗ |x∗(u−x0) +z∗(v−f(x0))≤0, ∀(u, v)∈Cepif(x0)}. Observe that
Cepi◦ f(x0) = {(x∗, z∗)∈X∗×Z∗| x∗(x−x0) +z∗(y−f(x0))≤0, ∀(x, y)∈epif} (17) and that, if (x∗, z∗)∈Cepi◦ f(x0), then z∗ ∈ −Z+∗.
If f takes the value −∞ and (x∗, z∗)∈Cepi◦ f(x), thenz∗ = 0. If in addition f(x)<+∞ for allx∈X, then x∗ = 0 and so, Cepi◦ f(x0) ={(0,0)}.
Unless otherwise stated, we will consider functions which are proper.
Definition 4.1 Let z∗ ∈ Z+∗ \ {0}, f : X → Z be a lower semicontinuous function, proper and x0 ∈domf. We note
∂z∗f(x0) ={T ∈ L(X, Z)|(z∗◦T,−z∗)∈Cepi◦ f(x0)} (18)
∂z∗f(x0) ={T ∈ L(X, Z)|(z∗◦T,−1)∈Cepi◦ z∗◦f(x0)} (19)
∂z∞∗ f(x0) ={T ∈ L(X, Z)|(z∗◦T,0)∈Cepi◦ z∗◦f(x0)} (20) and for z∗ ∈Z∗ \ {0}
∂z∞∗f(x0) ={T ∈ L(X, Z)|(z∗◦T,0) ∈Cepi◦ f(x0)}. (21) We use the following terminology: the set given by (18)(respectively (19),(20),(21)) is the convex subdifferential(respectively the scalar convex subdifferential, thesingular convex subdifferential and thesingular scalar convex subdifferential) of f at x0.
When Z = IR, and g : X →IR is a proper lower semicontinuous function then the two sets (27) and (28) coincide with the usual convex subdifferential given by
∂≤g(x0) ={x∗ ∈X∗ |x∗(x−x0)≤g(x)−g(x0),∀x∈X} (x0 ∈domg) while (30) and (29) coincide with the singular subdifferential given by
∂∞g(x0) ={x∗ ∈X∗ | x∗(x−x0)≤0, ∀x∈domg} (x0 ∈domg).
We remark that if domf =X, we can define∂z∗f(x0) and∂z∞∗ f(x0) for allz∗ ∈Z∗\{0}and these sets are nonempty if and only if∂≤z∗◦f(x) and∂∞z∗◦f(x) are nonempty respectively.
We make the observation that
z∗∈Z∗+\{0}
∂z∗f(x0) =∂≤f(x0), (22) where
∂≤f(x0) ={T ∈ L(X, Z) |T(x−x0)≤f(x)−f(x0), ∀x∈X} is the usual Fenchel subdifferential used for instance in [90]. If Z# =∅, then
∪
z∗∈Z#+
∂z∗f(x0)⊆∂>f(x0) where ∂>f(x0) ={T ∈ L(X, Z)| T(x−x0)> f(x)−f(x0)}.
Using the characterization of the normal cone Cepi◦ f(x0) we have:
T ∈∂z∗f(x0)⇐⇒z∗◦T(x−x0)≤z∗◦f(x)−z∗◦f(x0), ∀x∈X ⇐⇒z∗◦T ∈∂≤z∗◦f(x0).
It is a basic fact in convex analysis that
z∗◦T ∈∂≤z∗◦f(x0)⇐⇒(z∗◦T,−1)∈Cepi◦ z∗◦f(x0)⇐⇒T ∈∂z∗f(x0).
Thus, ∂z∗f(x0) =∂z∗f(x0).
In this case, the directional derivative, Df(x) is a closed convex process (by using the Rockafellar terminology) i.e.
Df(x)(λu) =λDf(x)(u), ∀λ≥0;
Df(x)(u) +Df(x)(v)⊆Df(x)(u+v);
Gr Df(x) is a closed set of Z.
Remark 4.1 Using the properties of Cepif(x), we get that for each u∈PrXCepif(x) ={u∈X | ∃v ∈Z, (u, v)∈Cepif(x)} we have:
Df(x)(u) +Z+ =Df(x)(u).
Proposition 4.1 Let f : X → Z be a proper lower semicontinuous function and z∗ ∈ Z+∗ \ {0}, x∈domf. Then we have:
cl Gr(z∗◦Df(x0)) = Gr (Dz∗◦f(x0)).
Proof. Since z∗ ∈Z+∗ \ {0}, (u, v)∈Cepif(x), implies that (u, z∗(v))∈Cepiz∗◦f(x) and thus cl Gr(z∗◦Df(x0))⊆Gr (Dz∗◦f(x0)).
Now, let (u, z∗(v)) ∈ Cepiz∗◦f(x0) = cl cone(epi z∗◦f −(x0, z∗ ◦f(x0)). This implies that there exist sequences (un)n∈IN and (vn)n∈IN such that un → u, vn → z∗(v) and (un, vn) ∈ cone(epi z∗ ◦f −(x0, z∗ ◦f(x0))). Let vn such that vn = z∗(vn) (we can take vn = evn
with z∗(e) = 1) and (un, vn) ∈ cone(epi f −(x0, f(x0)). Thus, vn ∈ Df(x0)(un). Hence, vn∈z∗◦Df(x0)(un) and finally, (u, v)∈cl Gr(z∗◦Df(x0)).
So, we have the converse inclusion
cl Gr(z∗◦Df(x0))⊇Gr (Dz∗◦f(x0))
and the equality follows. ✷
Corollary 4.1 Let f : X → Z be a proper lower semicontinuous function x ∈ dom f and z∗ ∈Z+∗ \ {0}. Then for all u∈PrXCepif(x) we have:
z∗ ◦Df(x)(u)⊆Dz∗◦f(x)(u). (23) Corollary 4.2 Let f :X →Z be a proper lower semicontinuous function and x0 ∈domf.
1. T ∈∂≤f(x0) if and only if T u≤v, ∀v ∈Df(x0)(u), ∀u∈PrXCepif(x0);
2. If f is convex and the interior of Z+ is nonempty then T ∈ w∂>f(x0) if and only if T u>K v, ∀v ∈Df(x0)(u), ∀u∈PrXCepif(x0).
Proposition 4.2 For x0 ∈domf, z∗ ∈Z+∗ \ {0}, the following equality does hold:
z∗◦∂z∗f(x0) =∂≤z∗◦f(x0).
Proof. The inclusionz∗◦∂z∗f(x0)⊆∂≤z∗◦f(x0) is ob vious.
Now, take x∗ ∈∂≤z∗◦f(x0), and T ∈ L(X, Z) such thatx∗(x) =z∗◦T(x). Thus, z∗◦T ∈
∂≤z∗◦f(x0), which amounts to saying thatT ∈∂z∗f(x0). Indeed, we can take the operator T given by T x = x∗(x)e with e an element of Z such that z∗(e) = 1. Since x∗ ∈ X∗ and z∗◦T =x∗, T ∈ L(X, Z) and thus, z∗◦∂z∗f(x0)⊇∂≤z∗◦f(x0) and the equality follows. ✷
We recall the definition of generalizated jacobians used in [60].
Definition 4.2 Letf :X →Z by a convex proper function and x∈domf. The multifunc-tion D∗f(x) :Z∗ −−→−→ X∗ given by
x∗ ∈D∗f(x)(z∗) ⇐⇒ (x∗,−z∗)∈Cepi◦ f(x) is called the adjoint jacobian.
We make the observation that for a lower semicontinuous function, for allz∗ ∈Z+∗ \ {0}, we have
D∗f(x)(z∗) =∂z∗◦f(x) =z∗◦∂z∗f(x).
If z∗ = 0, then
D∗f(x)(0) ={0}
z∗∈Z∗\{0}
z∗◦∂z∞∗f(x) We remark that for all z∗ ∈Z∗\ {0}, the sets
z∗◦∂z∞∗f(x)
are equal and thus we can write that for each z∗ ∈Z∗\ {0}, D∗f(x)(0) ={0}z∗◦∂z∞∗f(x).
Following the lines of the scalar case, we define now a type of vectorial conjugate which will be used for the characterization of the subdifferentiability.
Definition 4.3 Letf :X →Z be a proper lower semicontinuous function andT ∈ L(X, Z).
We note f∗(T) the vectorial conjugate of f atT given by
f∗(T) =SUP1{T x−f(x), x∈domf}.
Similarly, wf∗(T) is the weak vectorial conjugate of f at T which is defined by wf∗(T) =wSUP1{T x−f(x), x∈domf}.
Under the hypothesis of Theorem 1.3 we observe thatwf∗(T) = +∞orwf∗(T) is a nonemp-ty set inZ.
Proposition 4.3 Let f : X → Z be a proper lower semicontinuous function and x0 ∈ domf.
1. If z∗ ∈Z+∗ \ {0} and T ∈∂z∗f(x0), then
z∗(T x0−f(x0))≥maxz∗◦f∗(T).
2. If T ∈∂≤f(x0), then
T x0−f(x0) =f∗(T).
Proof. 1) Let z∗ ∈Z+∗ \ {0} and T ∈∂z∗f(x0). Then,
z∗(T x0−f(x0)) = max{z∗(T x−f(x)) |x∈domf}. Letα ∈f∗(x0). Thus, for all ε >0 there exists xε ∈X such that
T(xε)−f(xε)> α−ε.
Since z∗ ∈Z+∗ \ {0}, this yields:
max(z∗◦f∗(T))≤max{z∗(T x−f(x)) |x∈domf}. Finally, we obtain that
z∗(T x0−f(x0))≥maxz∗◦f∗(T).
2) If T ∈∂≤f(x0) then T x0−f(x0)≥T x−f(x) for allx ∈dom f and thus, T x0−f(x0)∈f∗(T).
Now, for α∈f∗(T) and ε >0 we findxε ∈X such that
α−ε < T xε−f(xε)< T x0−f(x0).
This implies that α ≤T x0−f(x0) and therefore we have T x0−f(x0) = f∗(T), as desired.
✷
We denote by w∂≤f(x0) and w∂>f(x0) the weak vectorial subdifferential and the weak Pareto subdifferentialrespectively, given by
w∂≤f(x0) ={T ∈ L(X, Z)| T x−T x0≤K f(x)−f(x0), ∀x∈domf}and w∂>f(x0) ={T ∈ L(X, Z)| T x−T x0 >K f(x)−f(x0), ∀x∈domf}, where K = IntZ+∪ {0}.
Proposition 4.4 Let f : X →Z be a proper lower semicontinuous function, x0 ∈ dom f, T ∈ L(X, Z) and suppose that the interior of the cone Z+ is nonempty.
1. If f is convex, then there exists z∗∈ Z+∗ \ {0} such that T ∈∂z∗f(x0) if and only if T x0−f(x0)∈wf∗(T);
2. If in addition Z+ is normal and Daniell, T ∈w∂≤f(x0) if and only if
T(x0)−f(x0) =wf∗(T). (24)
Proof. 1) First, suppose that there existsz∗ ∈Z+∗, z∗ = 0 such that T ∈∂z∗f(x0). Thus, z∗(T x0−f(x0)) = max{z∗(T x−f(x)) |x∈X}.
This yields that for each x∈domf
T x0−f(x0)∈/ T x−f(x)−Int Z+. (25) Otherwise, it would existz ∈Int Z+ such that z∗(z) = 0 which would imply thatz∗ = 0.
Thus, relation (24) gives that T x0−f(x0)∈wf∗(T).
Let consider now that T x0−f(x0)∈wf∗(T). Thus,
T x0−f(x0)∈ {/ T x−f(x)−IntZ+, | x∈domf}.
Since f is convex and the cone Z+ is convex, by the Hahn Banach separation theorem we can select z∗ ∈Z+∗ \ {0} such that
z∗(T x0−f(x0))≥z∗(T x−f(x)), ∀x∈domf.
Obviously, if f(x) = +∞, then the previous inequality does hold.
The last relation says exactly that T ∈∂z∗f(x0) and thus the assertion is proved.
2) The previous proposition yields that the condition is necessary and wf∗(T) = +∞. Suppose that (24) does hold. From Theorem 1.3 we know that for all x ∈ dom f, there is some α∈wf∗(T) such that
T x−f(x)−α∈ −K.
Since wf∗(T) = T x0−f(x0), we obtain that for allx∈domf T x−f(x)−(T x0−f(x0))∈ −K.
This amounts to saying that T ∈w∂≤f(x0). ✷
Proposition 4.5 Let f : X → Z be a proper, convex, lower semicontinuous function and suppose that IntZ+ =∅. Then, forx0 ∈domf,
w∂>f(x0) =
z∗∈Z+∗\{0}
∂z∗f(x0).
Proof. LetT ∈
z∗∈Z+∗\{0}∂z∗f(x0). This means that there exists z∗ ∈Z+∗ \ {0} such that:
z∗(T x−T x0)≤z∗(f(x)−f(x0)), ∀x∈domf.
Since the interior of the cone Z+ is nonempty we have
T x−T x0 >K f(x)−f(x0), ∀x∈domf. (26)
This amounts to saying that T ∈w∂>f(x0).
Suppose now thatT ∈w∂>f(x0) and thus (26) does hold. Sincef is convex, from the Hahn-Banach separation theorem applied to the setsT x0−f(x0) and{T x−f(x)−K | x∈dom f}, we can select z∗ ∈Z+∗ \ {0} such that
z∗(T x−T x0)≤z∗(f(x)−f(x0)), ∀x∈domf. (27) Relation (35) says exactly that T ∈∂z∗f(x0) and the proof is complete. ✷ We recall from [90] that ifX is a reflexive Banach space,Z a locally convex space ordered by a convex coneZ+ with the property that IntZ+∗ =∅, then for a convex functionf,∂≤f(x0) is a nonempty set at each pointx0 of continuity for f.
Example 1. If the order is not total, then we can easily give an example of a mapping f:X → Z. such that ∂≤f(0) = ∅ and ∂>f(0) = ∅ . For this, pick T ∈ L(X, Z) and define f(x) =T(x) +α (whereα∈Z\(Z+∪ −Z+)) for allx= 0 and f(0) = 0. Then, T ∈∂>f(0) while, ∂≤f(0) =∅.
Example 2. Obviously, the inclusion ∂≤f(x0)⊆∂>f(x0), holds for all x0 ∈dom f. LetX and Z be two Hausdorff locally convex spaces and f:X →Z a constant function. Then,
∂≤f(x0) ={0} and ∂>f(x0) ={T ∈ L(X, Z)|T(x)>0, T(x)<0, ∀x∈X}.
If the order is not total, then for x∗ ∈ X∗ \ {0} and α ∈ Z \ (Z+ ∪ −Z+), the operator T =αx∗ belongs to∂>f(x0) andT = 0. Thus, ∂≤f(x0)⊂ ∂>f(x0).
The following proposition provides a necessary and sufficient condition for the equality of these two types of vectorial subdifferentials.
Proposition 4.6 Let X and Z be two Hausdorff, locally convex spaces and f:X → Z a mapping such that ∂≤f(x0) = ∅. Then ∂≤f(x0) = ∂>f(x0) if and only if “≤” is a total ordering relation.
Proof. Suppose that “≤” is not a total order. Thus Z has an algebraic dimension larger or equal to 2 and it existsz ∈Z\(Z+∪Z−). The operatorT1∈ L(X, Z) given b yT1(x) =x∗(x)z with z ∈Z\(Z+∪ −Z+) and x∗ ∈X∗\ {0} has the property that
T1 = 0 and ∀x∈X, T1(x)>0, T1(x)<0.
Now for T in ∂≤f(x0) we have :
(T +λT1)(x−x0)> f(x)−f(x0), ∀x∈X, λ∈IR+.
Therefore,T+λT1 belongs to∂>f(x0) and since the subdifferentials are equal inx0, we have that T +λT1 belongs to∂≤f(x0) for all λ >0. This means that
(T +λT1)(x−x0)≤f(x)−f(x0), ∀x∈X, ∀λ >0.
In particular, for an arbitrarilyx we have
T1(x−x0)≤(f(x)−f(x0)−T(x−x0))/λ, ∀λ >0.
As a result, this yieldsT1(x−x0)≤0 for allxinX, which implies thatT1 = 0, a contradiction.
For the sufficiency, it is obvious that if the order is total, then the two subdifferentials
coincide. $
Proposition 4.7 LetX andZ be real reflexive Banach spaces and suppose that the ordering cone Z+ has a weakly compact base and has nonempty interior. Let f, g:X → Z be convex continuous functions. The following assumptions are equivalent:
1. ∂≤f(x)⊆∂≤g(x), for all x in X;
2. ∂≤f(x) =∂≤g(x), for all x in X;
3. f(x) =g(x) +k, for all x in X.
Proof. From [38] Lemma 2.26, pp. 52,
∂≤(z∗◦f)(x) = z∗◦∂≤f(x)⊆z∗◦∂≤g(x) = ∂≤(z∗◦g)(x).
Thus, ∂≤(z∗◦f)(x)⊆∂≤(z∗◦g)(x) which yields
z∗◦f(x) =z∗◦g(x) +k(z∗), ∀z∗∈ Z+∗, ∀x∈X.
In particular, z∗◦f(x)−z∗◦g(x) =z∗◦f(0)−z∗◦g(0), ∀z∗ ∈Z+∗. Since IntZ+∗ =∅, the cone Z+∗ generates Z∗, i.e. Z∗ =Z+∗ −Z+∗ and the relation (above) holds for allz∗ ∈Z∗. As a result,
f(x)−g(x) =f(0)−g(0) =k,
and the equivalences follow obviously. $