A Family of Enlargements of Maximal Monotone Operators
B.F. Svaiter ([email protected])∗
IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110.
Rio de Janeiro, RJ, CEP 22460-320, Brazil.
Abstract. We introduce a family of enlargements of maximal monotone operators.
The Brønsted and Rockafellar ε-subdifferential operator can be regarded as an
enlargement of the subdifferential. The family of enlargements introduced in this
paper generalizes the Brønsted and Rockafellar ε-subdifferential (enlargement) and
also generalize the enlargement of an arbitrary maximal monotone operator recently
proposed by Burachik, Iusem and Svaiter.
We characterize the biggest and the smallest enlargement belonging to this family
and discuss some general properties of its members.
A subfamily is also studied, namely, the subfamily of those enlargements which
are also additive. Members of this subfamily are formally closer to the ε-subdifferential.
Existence of maximal elements is proved. In the case of the subdifferential, we prove
that the ε-subdifferential is maximal in this subfamily.
Keywords: maximal monotone operators, enlargements, Banach spaces.
Mathematics Subject Classification (1991): 47H05, 46B99
1. Introduction and Motivation
Let A and B be arbitrary sets and F : A ⇒ B a multifunction. By an
enlargement or extension of F we mean a multifunction E : R+ ×A ⇒ B
such that
F (x) ⊆ E(b, x) ∀b > 0, x ∈ A.
A well known and most important example of extension of a multifunction is the ε-subdifferential. Given a proper convex function f on
a Banach space X, f : X → R ∪ {+∞}, the subdifferential of f at x,
i.e., the set of subgradients of f at x, denoted by ∂f (x), is given by
∂f (x) = {v ∈ X ∗ : f (y) > f (x) + hv, y − xi for all y ∈ X} .
It is trivial to check that ∂f is monotone. Rockafellar proved in [12] that
if f is also lower-semicontinuous, then ∂f is maximal monotone. The
ε-subdifferential operator was introduced by Brønsted and Rockafellar
in [1]. It is defined as
∂ε f (x) := {v ∈ X ∗ : f (y) > f (x) + hv, y − xi − ε for all y ∈ X} , (1)
∗
Partially supported by CNPq Grant 301200/93-9(RN) and by PRONEX–
Optimization.
c 2000 Kluwer Academic Publishers. Printed in the Netherlands.
fam-enlarg.tex; 1/01/2000; 1:09; p.1
2
for any ε > 0, x ∈ X. Note that ∂f = ∂0 f and ∂f (x) ⊆ ∂ε f (x) for any
ε > 0. So, we have an example of an enlargement of certain monotone
operators (those which are subdifferentials).
For an arbitrary monotone operator T : X ⇒ X ∗ , the following
enlargement was proposed in [5]: Given ε > 0 and x ∈ X
T ε (x) = {v ∈ X ∗ | hu − v, y − xi ≥ −ε, ∀y ∈ X, u ∈ T (y)},
(2)
This concept was originally presented for monotone operators in RN by
Burachik, Iusem and Svaiter in [2]. Subsequently, it was extended for
Hilbert spaces [3] and Banach spaces [5]. As for the ε-subdifferential,
T ⊆ T ε for any ε > 0 (follows from monotonicity of T ). Hence, T ε is
another example of enlargement of monotone operators.
If T is maximal monotone, we have another similarity with the
ε-subdifferential: T = T 0 (follows from maximal monotonicity). Furthermore, in this case, T ε shares with the ε-subdifferential of a proper
closed convex function many properties, e.g., local boundedness, demiclosed graph, a transportation formula, Lipschitz continuity, regardless
of T being or not a subdifferential [3, 5]. Additionally, if X is reflexive,
T ε satisfies also a Brønsted & Rockafellar property [5]. Natural questions are: why is that T ε and ∂ε f share these properties? which are
the enlargements of an arbitrary maximal monotone operator which
share with the T ε enlargement and with the ε-subdifferential those
properties?
The T ε enlargement does not coincide in general with the ε-subdifferential when T = ∂f [2]. This rises another question: Is there another
kind of enlargement defined for arbitrary maximal monotone operators
which is formally closer to the ε-subdifferential enlargement?
In this paper, we introduce a family of enlargements of maximal
monotone operators, which includes T ε as a particular element. In the
case in which T is a subdifferential, the ε-subdifferential also belongs
to this family.
Enlargements belonging to this family share with the ε-subdifferential
and with T ε many properties. We characterize the biggest and the
smallest of these enlargements. We also define a subfamily whose members are formally closer to the ε-subdifferential. This is the subfamily of
enlargements which are also additive. We prove that the ε-subdifferential
is maximal in the “additive” subfamily of enlargements of the subdifferential. We prove that maximal monotone operators always have
maximal enlargements in the associated “additive” subfamily.
This paper is organized as follows. In Section 2 we give some basic
definitions. In Section 3 we define formally the new family of enlargements and find its extrema. In Section 4 we discuss some general
properties of the elements of this family. In Section 5 we further an-
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3
A Family of Enlargements
alyze the smallest enlargement belonging to the family. In Section 6
we introduce the notion of additive enlargements. We prove that the
subfamily of additive enlargements of an arbitrary maximal monotone
operator is nonempty and has maximal elements. In the particular case
of a subdifferential, the ε-subdifferential is maximal in the additive
subfamily. In Section 7 we discuss some operations with enlargements.
2. Basic definitions
From now on X is a real Banach space, X ∗ is its dual. Given x ∈ X
and v ∈ X ∗ , v(x) will be denoted indifferently by hx, vi and hv, xi. In
products of Banach spaces (e.g., X ×X ∗ , R×X ×X ∗ ) we shall consider
the canonical product topology.
Given a multifunction F : X ⇒ X ∗ ,
− for A ⊆ X, F (A) is defined as F (A) :=
S
a∈A F (a).
− F is locally bounded at x if there exists a neighborhood U of x such
that the set F (U ) is bounded.
− The domain, image and graph of F are respectively denoted by
D(F ) := {x ∈ X : F (x) 6= ∅} ,
R(F ) := F (X) and
G(F ) := {(x, v) : x ∈ X and v ∈ F (x)} .
− The closure of F , F̄ : X ⇒ X ∗ is defined as
F̄ (x) := {v ∈ X ∗ : (x, v) ∈ G(F )}.
F is closed if F = F̄ (or equivalently, if G(F ) is closed).
− F is monotone if hu − v, x − yi ≥ 0 for all u ∈ F (x) and v ∈ F (y),
for all x, y ∈ X.
− F is maximal monotone if it is monotone and its graph is maximal
with respect to this property, i.e., it is not properly contained in
the graph of any other monotone operator.
The notation above is fairly standard. We will need some additional
notation, to deal with enlargements.
Given E : R+ × X ⇒ X ∗
− The graph of E, G(E) is
G(E) := {(b, x, v) ∈ R × X × X ∗ | v ∈ E(b, x)}.
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4
− The closure of E, E : R+ × X ⇒ X ∗ is defined as
E(b, x) := {v ∈ X ∗ : (b, x, v) ∈ G(E)}.
E is closed if E = E (or equivalently, if G(E) is closed).
− We say that E is an enlargement of F : X ⇒ X ∗ if for any x ∈ X,
b>0
F (x) ⊆ E(b, x).
Given f a proper lower-semicontinuous convex function on X, the
ε-subdifferential of f (as a multifunction on ε and x ∈ X), R+ × X 3
(ε, x) 7→ ∂ε f (x) will be denoted by ∂f , i.e.,
∂f : R+ × X ⇒ X ∗ ,
∂f (b, x) := ∂b f (x).
(3)
We will call ∂f the Brønsted-Rockafellar enlargement of ∂f .
We are concerned with enlargements of monotone operators (indeed,
maximal monotone operators). Since ε is associated with many different
concepts, we will change our notation for the T ε enlargement. From
now on this enlargement will be called B-enlargement, and denoted by
B T . So, given T : X ⇒ X ∗ monotone, the B-enlargement of T is the
application B T : R+ × X ⇒ X ∗ ,
B T (b, x) := T b (x)
= {v ∈ X ∗ | hv − u, x − yi > −b, ∀y, u ∈ T (y)}
(4)
3. A New Family of Enlargements of Maximal Monotone
operators
Although some of the considerations below could be applied to monotone multifunctions, we are concerned in this paper with maximal
monotone multifunctions. Therefore, from now on,
T : X ⇒ X∗
is an arbitrary maximal monotone operator.
DEFINITION 3.1. Denote by ∆n the unit simplex of Rn :
(
∆n :=
n
α ∈ R | α > 0,
n
X
)
αi = 1 .
i=1
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A Family of Enlargements
5
We say that E : R+ × X ⇒ X ∗ satisfies the transportation formula if
for any n ∈ N, given n points (bi , xi , v i ) ∈ G(E) and some α ∈ ∆n ,
defining
n
X
x=
b=
i=1
n
X
αi xi , v =
n
X
αi v i ,
(5)
i=1
αi bi + αi hv i − v, xi − xi
(6)
i=1
it holds that (b, x, v) ∈ G(E) (i.e., b > 0 and v ∈ E(b, x)).
As α ∈ ∆n , using also (5),
Pn
i=1 αi hv
i
− v, xi − xi =
=
Pn
P
P
α hv i , xi i − hv, ni=1 αi xi i − h ni=1 αi v i , xi
i=1
Pn i
+
α hv, xi
Pn i=1 ii i
i=1 αi hv
, x i − hx, vi.
Hence, in this definition, equation (6) is equivalent to
"
b=
n
X
#
i
i
αi (bi + hv , x i) − hv, xi.
(7)
i=1
The same definition has been presented in [3, 5]. The B T enlargement and the ε-subdifferential satisfy the transportation formula. This
property was essential in proving Lipschitz continuity of B T [3] and for
some algorithmic applications of this enlargement [3, 4, 13].
LEMMA 3.2. Define
Ψ : R × X × X ∗ → R × X × X ∗,
Ψ(b, x, v) := (b + hx, vi, x, v).
Take E : R+ × X ⇒ X ∗ . Then E satisfies the transportation formula
if and only if Ψ(G(E)) is convex.
Proof. Observe that Ψ is a bijection. Take n pairs of corresponding
points (bi , xi , v i ) ∈ G(E), (ti , xi , v i ) ∈ Ψ(G(E)) (i = 1, . . . , n). Then
ti = bi + hxi , v i i,
i = 1, . . . , n.
(8)
Take some α ∈ ∆n and define
x=
b=
n
X
αi xi ,
i=1
" n
X
v=
n
X
αi v i ,
(9)
i=1
#
i
i
αi (bi + hx , v i) − hx, vi.
(10)
i=1
fam-enlarg.tex; 1/01/2000; 1:09; p.5
6
Note that from (10), (8), it follows that
"
b=
n
X
#
αi ti − hx, vi.
(11)
i=1
Therefore, using also (9) we get
n
X
αi (ti , xi , v i ) = (b + hx, vi, x, v).
(12)
i=1
Suppose first that E satisfies the transportation formula. Then,
using (9), (10) we conclude that (b, x, v) ∈ G(E). Therefore,
(b + hx, vi, x, v) ∈ Ψ(G(E)).
Using also (12) we conclude that
n
X
αi (ti , xi , v i ) ∈ Ψ(G(E)).
i=1
Since the points (ti , xi , v i ) may be chosen arbitrarily in Ψ(G(E)) (and
α is an arbitrary element of ∆n ) we conclude that Ψ(G(E)) is convex.
Suppose now that Ψ(G(E)) is convex. Then,
n
X
αi (ti , xi , v i ) ∈ Ψ(G(E)).
i=1
Using also (12) we obtain
(b + hx, vi, x, v) ∈ Ψ(G(E)).
Therefore
(b, x, v) = Ψ−1 (b + hx, vi, x, v) ∈ G(E).
Since the points (bi , xi , v i ) may be chosen arbitrarily in G(E) (and
α is an arbitrary element of ∆n ) we conclude that E satisfies the
transportation formula.
2
DEFINITION 3.3. Let IE(T ) be the family of enlargements of T , E :
R+ × X ⇒ X ∗ with the following properties:
(p1) 0 6 b1 6 b2 ⇒ E(b1 , x) ⊆ E(b2 , x) for any x ∈ X, b1 , b2 ∈ R+ .
(p2) E satisfies the transportation formula.
LEMMA 3.4.
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A Family of Enlargements
7
1. B T ∈ IE(T ). In particular, IE(T ) 6= ∅.
2. Let f be a lower semicontinuous proper convex function on X.
Then the ε-subdifferential enlargement of ∂f belongs to IE(∂f ), i.e.,
∂f ∈ IE(∂f ).
Proof. To prove item 1, first recall that T is maximal monotone,
hence monotone. From (4) it follows trivially that B T is an enlargement
of T . Property (p1) for B T follows directly from (4) again. Property
(p2) for B T is proved in [5, Theorem 3.11].
To prove item 2, first observe that from (1) it follows that ∂f is
an enlargement of ∂f . Property (p1) also follows directly from (1).
Property (p2) for ∂f is proved in [8, Proposition 1.2.10].
2
In the particular case in which T is a subdifferential of some function
f , i.e., T = ∂f , is very natural to investigate the relations between the
ε-subdifferential enlargement of ∂f = T and B T = B ∂f . This was done
in [2, Proposition 3], in a finite dimensional setting, where is proved
that for any ε > 0:
∂ε f (x) ⊆ B ∂f (ε, x),
which is equivalent to
G(∂f ) ⊆ G(B ∂f ).
The extension of the proof for a Banach Space is trivial. Still in [2] an
example is given where the above inclusion is proper. We shall prove a
more general result.
In the family of enlargements we will consider the (partial) order
of inclusion of the graphs. Hence, if E1 and E2 are enlargements of
T , we say that E1 is smaller than E2 (or E2 is bigger than E1 ) if
G(E1 ) ⊆ G(E2 ).
THEOREM 3.5. The enlargement B T is the biggest element of IE(T ).
Proof. Suppose that there exist E ∈ IE(T ) which is not smaller than
B T (in the partial order of inclusion of the graphs): G(E) 6⊆ G(B T ).
Then there exist
(c, z, w) ∈ G(E)
(13)
which is not in G(B T ), i.e., w ∈
/ B T (c, z). From (4) it follows that there
exist y ∈ X, u ∈ T (y) such that
hw − u, z − yi < −c.
(14)
Since E is an enlargement of T and u ∈ T (y),
(0, y, u) ∈ G(E).
(15)
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8
Take some θ ∈ (0, 1) and define
α1 := θ,
α2 := 1 − θ.
Observe that (α1 , α2 ) ∈ ∆2 . Using the transportation formula with
α = (α1 , α2 ) and the points
(b1 , x1 , v 1 ) = (c, z, w) ∈ G(E),
(b2 , x2 , v 2 ) = (0, y, u) ∈ G(E)
we conclude that for (b, x, v) given by
x := α1 z + α2 y, v := α1 w + α2 u,
b := α1 c + α1 hz − x, w − vi + α2 hy − x, u − vi
it holds that
b > 0,
(b, x, v) ∈ G(E).
Direct calculation yields
b = θc + θ(1 − θ)hw − u, z − yi
= θ [c + (1 − θ)hw − u, z − yi] .
Therefore (recall that θ > 0)
c + (1 − θ)hw − u, z − yi > 0.
Since θ is an arbitrary number in (0, 1), we conclude that
c + hw − u, z − yi > 0,
2
in contradiction with (14).
Once we have the biggest enlargement in the family, let us go to the
smallest.
LEMMA 3.6. The family IE(T ) has a smallest element, given by
S T : R+ × X ⇒ X ∗ ,
(16)
T
S T (b, x) := E∈IE(T ) E(b, x).
Proof. First observe that IE(T ) is nonempty. So, S T , as given in (16),
is well defined and is also an enlargement of T . Since any E ∈ IE(T )
satisfies conditions (p1) and (p2) of Definition 3.3, it follows that S T
also satisfies these conditions. So S T ∈ IE(T ) and G(S T ) ⊆ G(E) for
any E ∈ IE(T ).
2
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A Family of Enlargements
9
4. General Properties
As we said in Section 1, enlargements in the new family (Definition 3.3)
share with the ε-subdifferential enlargement many properties.
LEMMA 4.1. Take E ∈ IE(T ). Then
1. For any x ∈ X, E(0, x) = ∩b>0 E(b, x) = T (x).
2. For any x ∈ X, b ∈ R, the set E(b, x) is convex.
3. For any v ∈ X ∗ , b ∈ R, the set
{x ∈ X | v ∈ E(b, x)}
is convex.
Proof. To prove item 1 observe that since E is an enlargement
of T , T (x) ⊆ E(0, x). From (p1) of Definition 3.3 it follows that
E(x, 0) ⊆ ∩b>0 E(b, x). From Theorem 3.5 it follows that ∩b>0 E(b, x) ⊆
∩b>0 B T (b, x). Definition (4) gives ∩b>0 B(b, x) = B(0, x). Since T is
maximal monotone, from (4) it follows that B T (0, x) = T (x) and the
conclusion follows.
To prove item 2 take v 1 , v 2 ∈ E(b, x) and p, q > 0, p+q = 1. Applying
the transportation formula we conclude that pv 1 + qv 2 ∈ E(b, x). The
proof of item 3 is analogous to the proof of item 2.
2
Recall that an enlargement is closed if it has a closed graph. One
may easily check that the graph of the ε-subdifferential and of the B T
enlargement are closed. Hence these are examples of closed enlargements in IE(∂f ) and IE(T ) respectively. Indeed, for these enlargements
a stronger property holds, they have a demiclosed graph. We will state
this formally only for the B T enlargement:
PROPOSITION 4.2 ([5, Proposition 3.4]). The graph of B T is demiclosed, i.e.,
(a) If {xk } ⊂ X converges strongly to x0 , {uk ∈ B T (bk , xk )} converges weak ∗ to u0 in X ∗ and {bk } ⊂ R+ converges to b0 , then
u0 ∈ B T (b0 , x0 ).
(b) If {xk } ⊂ X converges weakly to x0 , {uk ∈ B T (bk , xk )} converges strongly to u0 in X ∗ and {bk } ⊂ R+ converges to b0 , then
u0 ∈ B T (b0 , x0 ).
This property is shared in a smaller degree by closed enlargements in
IE(T ).
fam-enlarg.tex; 1/01/2000; 1:09; p.9
10
PROPOSITION 4.3. Take E ∈ IE(T ) closed. Then
(a) If {xk } ⊂ X converges strongly to x0 , {uk ∈ E(bk , xk )} converges weakly to u0 in X ∗ and {bk } ⊂ R converges to b0 , then
u0 ∈ E(b0 , x0 ).
(b) If {xk } ⊂ X converges weakly to x0 , {uk ∈ E(bk , xk )} converges
strongly to u0 in X ∗ and {bk } ⊂ R converges to b0 , then u0 ∈
E(b0 , x0 ).
Proof. To prove item (a), it is enough to show that (b0 , x0 , u0 ) is
in the closure of G(E). Since {uk } converges weakly, it is bounded, so
there exist some M > 0 such that
kui k 6 M.
∀i ∈ N,
In particular, ku0 k 6 M .
Now take some arbitrary ε > 0. There exist some N ∈ N such that
i > N ⇒ |bi − b0 | 6 ε, kxi − x0 k 6 ε.
Note that u0 is in the closure of the convex hull of {uk }k>N . Hence,
there exist some n ∈ N and αn ∈ ∆n such that,
n
X
N +i
0
αi u
− u 6 ε.
i=0
Now, define
x=
n
X
αi xN +i ,
b=
n
X
n
X
αi uN +i
i=0
i=0
and
u=
αi bN +i + αi hxN +i − x, uN +i − ui.
i=0
It follows, from transportation formula that
(b, x, u) ∈ G(E).
We already know that ku − u0 k 6 ε. Note also that
kx − x0 k 6
n
X
αi kxN +i − x0 k 6 ε
i=0
and
P
n
|b − b0 | = N +i − x, uN +i − ui − b0
i=0 αi bN +i + αi hx
P
6 |( ni=0 αi bN +i ) − b0 | + ni=0 αi hxN +i − x, uN +i − ui
P
P
6 n0 αi |bN +i − b0 | + ni=0 αi kxN +i − xkkuN +i − uk
P
6 ε + 2M ni=0 αi kxN +i − xk
P
6 ε + 2M ni=0 αi (kxN +i − x0 k + kx0 − xk)
6 ε + 4M ε.
P
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A Family of Enlargements
11
Since ε is arbitrary, it follows that (b0 , x0 , u0 ) is in the closure of G(E).
The proof of item (b) is analogous.
2
It is a well-known fact (see [7], [11]) that any maximal monotone
operator in an arbitrary Banach space is locally bounded on the interior
of its domain. Enlargements in the family IE inherit this property, the
bound being a linear (affine) function on the parameter b.
PROPOSITION 4.4 (“affine” local boundedness). For any x in the interior of D(T ) there exist a neighborhood of x, U ⊂ int D(T ) and
constants L, M > 0, such that
sup{kvk | x ∈ U, v ∈ E(b, x)} 6 bL + M
for any E ∈ IE(T ), b > 0.
Proof. From Theorem 3.5 it follows that E(b, x) ⊆ B T (b, x). Since
T
B is “affine” local bounded [5, Corollary 3.10], the conclusion follows.
2
Closeness between bounded subsets of X ∗ can be measured using
the Hausdorff distance. For v ∈ X ∗ and W ⊆ X ∗ (nonempty), define
d(v, W ) := inf{w ∈ W | kv − wk}.
Then, given V, W ⊆ X ∗ nonempty bounded subsets of X ∗ ,
dH (V, W ) := max{sup d(v, W ), sup d(w, V )}.
v∈V
w∈W
Observe that if additionally, V, W are closed, then dH (V, W ) = 0 is
equivalent to V = W .
With this construct, it is natural to question whether a given pointto-set operator is continuous. The subdifferential of a proper lowersemicontiunous function is not continuous in this extended distance.
Surprisingly, the ε-subdifferential is continuous and even Lipschitz continuous, for ε > 0. This result (Lipschitz continuity) was first obtained
by Nurminskii for a fixed ε [10]. Latter on, it was improved by HiriartUruty [6]. This property is a very nice feature of the ε-subdifferential.
In some sense it shows that this enlargement regularizes (or smooths)
the subdifferential. When an enlargement turns a possibly discontinuous multifunction into a Lipschitz continuous (enlarged) multifunction,
we say that the enlargement has the Nurminskii Property. So, the
Brøndsted-Rockafellar enlargement of the subdifferential has the Nurminskii Property. The same holds for the B-enlargement [5]. Indeed,
Nurminskii Property is a common feature in the family IE.
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12
PROPOSITION 4.5. Let K be a compact set contained in the interior
of D(T ) and 0 < b 6 b < +∞. There exists an open set V ,
K ⊂ V ⊂ D(T )
and nonnegative constants A and B such that for any E ∈ IE(T ) and
(b1 , x1 ), (b2 , x2 ) ∈ [b, b] × V
dH (E(b1 , x1 ), E(b2 , x2 )) 6 Akx1 − x2 k + B|b1 − b2 |.
Proof. Use Theorem 3.5 and Definition 3.3 to conclude that conditions (E1-E3) of [5, Section 3.4] are satisfied. Then apply [5, Theorem
3.14, Corollary 3.15]. The constants A, B can be chosen regardless of
the particular E ∈ IE(T ).
2
Now we will discuss a property of the ε-subdifferential that is not
shared by enlargements in IE(T ), in general. Let f be a proper closed
convex function on X:
If v ∈ ∂ε f (x), v 0 ∈ ∂ε0 f (x0 ) then
hv − v 0 , x − x0 i > −(ε + ε0 ).
(17)
This result follows directly from (1). For B T a weaker result holds [5,
Corollary 3.12]:
If v ∈ B T (b, x), v 0 ∈ B T (b0 , x0 ) then
√
√ 2
b + b0 .
hv − v 0 , x − x0 i > −
(18)
Since B T is the biggest enlargement in IE(T ), this result extends to
any E ∈ IE(T ).
5. Study of S T
We shall get another characterization of S T , which will be useful in the
next section.
DEFINITION 5.1. Let CT be the set of those (b, x, v) ∈ R × X ×
X ∗ such that there exist α ∈ ∆n and (xi , v i ) ∈ G(T ), (i = 1, . . . , n)
satisfying:
P
P
x = ni=1 αi xi , v = ni=1 αi v i ,
(19)
P
b > ni=1 αi hxi − x, v i − vi.
We will prove that G(S T ) = CT . First we claim that
CT ⊆ G(S T ).
(20)
fam-enlarg.tex; 1/01/2000; 1:09; p.12
13
A Family of Enlargements
To prove it, take (b, x, v) ∈ CT . Then there exists α ∈ ∆n and (xi , v i ) ∈
G(T ) satisfying (19). Observe that (0, xi , v i ) ∈ G(S T ), and S T satisfies
the transportation formula. So, defining
b̃ :=
n
X
αi hxi − x, v i − vi
i=1
it follows that v ∈ S T (b̃, x) and b > b̃. Using also property (p1) of
Definition 3.3 we conclude that v ∈ S T (b, x). Hence (b, x, v) ∈ G(S T ),
which establishes the claim.
Observe that from (20) it follows that CT ⊆ R+ × X × X ∗ . Now
define
S : R+ × X ⇒ X ∗ ,
(21)
S(b, x) := {v ∈ X ∗ | (b, x, v) ∈ CT }.
Then,
G(S) = CT .
(22)
Take v ∈ T (x). Choosing n = 1, α = 1 and (x1 , v 1 ) = (x, v) in
Definition 5.1 we conclude that (0, x, v) ∈ CT . Hence
T (x) ⊆ S(0, x),
∀x ∈ X.
(23)
Furthermore, directly from Definition 5.1 and (21) it follows that
0 6 b1 6 b2 ⇒ S(b1 , x) ⊆ S(b2 , x), ∀x ∈ X.
(24)
There is still one condition missing to conclude that S ∈ IE(T ).
LEMMA 5.2. The multifunction S (defined in (22)) satisfies the transportation formula.
Proof. Let Ψ be as in Lemma 3.2. Using Definition 5.1 it follows that
Ψ(CT ) is the set of those (t, x, v) such that there exist α ∈ ∆n and n
points (xi , v i ) ∈ G(T ) such that
x=
t>
Pn
i
i
i=1 αi x , v =
i=1 αi v ,
Pn
i
i
i=1 αi hx − x, v − vi + hx, vi
Pn
=
Pn
i i
i=1 αi hx , v i.
Therefore, Ψ(CT ) is the convex hull of
{(t, x, v) ∈ R × X × X ∗ | t > hx, vi, (x, v) ∈ G(T )}.
In particular, Ψ(CT ) is convex. From (22) it follows trivially that Ψ(G(S)) =
Ψ(CT ). Using Lemma 3.2 the conclusion follows.
2
THEOREM 5.3. G(S T ) = CT .
fam-enlarg.tex; 1/01/2000; 1:09; p.13
14
Proof. Take S defined in (21). From (23), (24) and Lemma 5.2 it
follows that S ∈ IE(T ). Therefore G(S T ) ⊆ G(S). Using also (22) and
(20) the conclusion follows.
2
6. Additive enlargements
In Section 3 we proved that enlargements in IE(T ) share with the εsubdifferential enlargement some properties. Property (18) is weaker
than (17). So we have an example of a property of the ε-subdifferential
which is not shared by a generic elements of IE(T ). This will be the
key for defining a subfamily of IE(T ) which is formally closer to the
ε-subdifferential.
DEFINITION 6.1. We say that E : R+ × X ⇒ X ∗ is additive if for
any (b1 , x1 , v 1 ), (b2 , x2 , v 2 ) ∈ G(E),
hv 1 − v 2 , x1 − x2 i > −(b1 + b2 ).
The subfamily of additive enlargements in IE(T ) will be denoted by
IE a (T ).
The Brønsted Rockafellar ε-subdifferential enlargement is additive
(follow from(17)). If the generic maximal monotone multifunction T
admits an additive enlargement in IE(T ), then, S T (being the smallest
element of IE(T )) shall be additive. Before verifying “additivenes” of
this enlargement, let us prove an interesting relation between S T and
BT .
LEMMA 6.2. For any v ∈ S T (b, x), v 0 ∈ B T (b0 , x0 ),
hv − v 0 , x − x0 i > −(b + b0 ).
Proof. Take v ∈ S T (b, x), v 0 ∈ B T (b0 , x0 ). Then, using Theorem 5.3
and Definition 5.1, we conclude that there exist α ∈ ∆n , and (xi , v i ) ∈
G(T ), i = 1, . . . , n satisfying
x=
b>
n
X
i=1
n
X
αi xi , v =
n
X
αi v i ,
(25)
i=1
αi hxi − x, v i − vi.
(26)
i=1
fam-enlarg.tex; 1/01/2000; 1:09; p.14
15
A Family of Enlargements
Then
hv − v 0 , x − x0 i =
=
n
X
i=1
n
X
αi hv i − v 0 , x − x0 i
h
i
αi hv i − v 0 , x − xi i + hv i − v 0 , xi − x0 i .
i=1
Since v 0 ∈ B T (b0 , x0 ) and v i ∈ T (xi ) (i = 1, . . . , n), from (4) it follows
that hv i − v 0 , xi − x0 i > −b0 . Therefore
0
0
0
hv − v , x − x i > −b +
n
X
αi hv i − v 0 , x − xi i.
i=1
Observe that
Pn
n
X
i=1 αi (x
− xi ) = 0. Hence,
αi hv i − v 0 , x − xi i =
i=1
n
X
αi hv i − v, x − xi i
i=1
> −b
where (26) was also used. Combining the two above inequalities, the
desired result follows.
2
PROPOSITION 6.3.
1. S T ∈ IE a (T ). In particular IE a (T ) 6= ∅.
2. Let f be a proper closed convex function on X. Then ∂f is additive.
So, ∂f ∈ IE a (∂f ).
Proof. We shall prove only item 1. Take v 1 ∈ S T (b1 , x1 ), v 2 ∈
S T (b2 , x2 ). From Lemma 3.4-item 1 and Lemma 3.6 it follows that
v 2 ∈ B T (b2 , x2 ). Now apply Lemma 6.2.
2
Next we will prove that Brønsted-Rockafellar enlargement is “maximal” in the subfamily of additive enlargements of the subdifferential
which satisfies also Definition 3.3. Recall that Fenchel-Moreau conjugate of f : X → R̄ is the function f ∗ : X ∗ → R̄,
f ∗ (v) := sup{hv, xi − f (x) | x ∈ X}.
If f is a proper closed convex function, then f ∗ is also a proper closed
convex function. The effective domain of f is the set where f < +∞:
ed(f ) := {x ∈ X | f (x) < +∞}.
fam-enlarg.tex; 1/01/2000; 1:09; p.15
16
THEOREM 6.4. Let f be a proper closed convex function on X. Then
∂f is maximal in IE a (∂f ). In other words, if E ∈ IE a (∂f ) and G(∂f ) ⊆
G(E) then E = ∂f .
Proof. Suppose that E ∈ IE a (T ) and G(∂f ) ⊆ G(E). Take (b0 , x0 , v 0 ) ∈
G(E). Then
hx0 − x, v 0 − vi > −(b0 + b),
∀(b, x, v) ∈ G(∂f ).
(27)
Observe that [1, eq. (3.1)]
v ∈ ∂b f (x) ⇔ b > f (x) + f ∗ (v) − hx, vi.
Therefore, taking in (27) x ∈ ed(f ), v ∈ ed(f ∗ ) and b = f (x) + f ∗ (v) −
hx, vi we get
hx0 −x, v 0 −vi > −(b0 +f (x)+f ∗ (v)−hx, vi),
∀x ∈ ed(f ), v ∈ ed(f ∗ ).
After rearranging we get
b0 > hv 0 , xi − f (x) + hv, x0 i − f ∗ (v) − hv 0 , x0 i,
∀x ∈ ed(f ), v ∈ ed(f ∗ ).
Observe that
f ∗ (v 0 ) = sup{hx, v 0 i − f (x) | x ∈ ed(f )},
f ∗∗ (x0 ) = sup{hv, x0 i − f ∗ (v) | v ∈ ed(f ∗ )}
so we conclude that
b0 > f ∗ (v 0 ) + f ∗∗ (x0 ) − hv 0 , x0 i.
Since f ∗∗ coincides with f in X [9], we conclude that
b0 > f ∗ (v 0 ) + f (x0 ) − hv 0 , x0 i
which is equivalent to v 0 ∈ ∂b0 f (x0 ). So (b0 , x0 , v 0 ) ∈ G(∂f ).
2
THEOREM 6.5. Given E ∈ IE a (T ) there exist an Ẽ maximal on
IE a (T ) and satisfying G(E) ⊆ G(Ẽ).
Proof. Follows directly from Zorn’s Lemma. Consider in IE a (T ) the
(partial) order of inclusion of the graphs. Let C be a totally ordered
subset of IE a (T ). Define F : R+ × X ⇒ X ∗ by
F (b, x) := ∪E∈C E(b, x).
Then F ∈ IE a (T ) and F is an upper bound for all elements of C.
Observe that
2
E ∈ IE(T ) ⇒ Ē ∈ IE(T ),
E ∈ IE a (T ) ⇒ Ē ∈ IE a (T ).
Hence
fam-enlarg.tex; 1/01/2000; 1:09; p.16
17
A Family of Enlargements
LEMMA 6.6. If E is maximal on IE a (T ) then E is closed.
7. Operations with enlargements
Let λ > 0. The operator λT is maximal monotone. Enlargements of λT
may be obtained trivially from the enlargements of T .
PROPOSITION 7.1. Take E ∈ IE(T ). Define λ E : R+ × X ⇒ X ∗ ,
λ E(b, x) := λE(λ−1 b, x).
Then λ E ∈ IE(λT ). Furthermore, if E is additive, λ E is also
additive.
Now let us discuss ‘addition’ of enlargements. Given E1 , E2 : R+ ×
X ⇒ X ∗ , define E1 ⊕ E2 : R+ × X ⇒ X ∗ as
E1 ⊕ E2 (b, x) :=
[
E1 (b1 , x) + E2 (b2 , x).
(28)
b1 ,b2 >0
b1 +b2 =b
LEMMA 7.2. Suppose that T1 , T2 : X ⇒ X ∗ are maximal monotone
operators. Suppose also that T1 + T2 is maximal monotone.
Take E1 ∈ IE(T1 ), E2 ∈ IE(T2 ). Let
E := E1 ⊕ E2 .
Then E ∈ IE(T1 + T2 ). Furthermore, if E1 and E2 are additive, then E
is additive.
Proof. From (28) and Definition 3.3, it follows trivially that E is an
enlargement of T1 + T2 and satisfies condition (p1) of Definition 3.3.
We shall prove that E satisfies the transportation formula.
Let Ψ be as in Lemma 3.2. From (28) it follows that
Ψ(G(E))
)
v 1 ∈ E (b , x),
1 1
=
(b1 + b2 + hx, v + v i, x, v + v ) 2
v ∈ E2 (b2 , x)
x∈X b1 ,b2 >0
(
)
1
[ [
1
2
1
2 v ∈ E1 (b1 , x),
=
(b1 + hx, v i + b2 + hx, v i, x, v + v ) 2
v ∈ E2 (b2 , x)
x∈X b1 ,b2 >0
(
)
(t , x, v 1 ) ∈ Ψ(G(E )),
[
1
1
=
(t1 + t2 , x, v 1 + v 2 ) .
(29)
(t2 , x, v 2 ) ∈ Ψ(G(E2 ))
(
[
[
1
2
1
2
x∈X
fam-enlarg.tex; 1/01/2000; 1:09; p.17
18
Now take (t, x, v), (t0 , x0 , v 0 ) ∈ Ψ(G(E)), p, q > 0, p + q = 1. According
to (29) there exist t1 , t2 ∈ R, v 1 , v 2 ∈ X ∗ ,
(t1 , x, v 1 ) ∈ Ψ(G(E1 )), (t2 , x, v 2 ) ∈ Ψ(G(E2 )),
t = t1 + t2 , v = v 1 + v 2
and t01 , t02 ∈ R, v 0 1 , v 0 2 ∈ X ∗ ,
(t01 , x0 , v 0 1 ) ∈ Ψ(G(E1 )), (t02 , x0 , v 0 2 ) ∈ Ψ(G(E2 )),
t0 = t01 + t02 , v 0 = v 0 1 + v 0 2 .
Observe that Ψ(G(E2 )), Ψ(G(E2 )) are convex. So
p(t1 , x, v 1 ) + q(t01 , x0 , v 0 1 ) = (pt1 + qt01 , px + qx0 , pv 1 + qv 0 1 ) ∈ Ψ(G(E1 )),
p(t2 , x, v 2 ) + q(t02 , x0 , v 0 2 ) = (pt2 + qt02 , px + qx0 , pv 2 + qv 0 2 ) ∈ Ψ(G(E2 )).
Using (29) again we conclude that
1
2
(p(t1 + t2 ) + q(t01 + t02 ), px + qx0 , p(v 1 + v 2 ) + q(v 0 + v 0 ))
= (pt + qt0 , px + qx0 , pv + qv 0 ) ∈ G(Ψ(E)).
Therefore, Ψ(G(E)) is convex and from Lemma 3.2 it follows that E
satisfies the transportation formula, hence E ∈ IE(T1 + T2 ).
Suppose now that E1 , E2 are also additive. Take (b, x, v), (b0 , x0 , v 0 ) ∈
G(E). Then there exist b1 , b2 > 0, v 1 , v 2 ∈ X ∗ such that
(b1 , x, v 1 ) ∈ G(E1 ), (b2 , x, v 2 ) ∈ G(E2 ),
b1 + b2 = b, v 1 + v 2 = v.
and b01 , b02 > 0, v 0 1 , v 0 2 ∈ X ∗ such that
(b01 , x0 , v 0 1 ) ∈ G(E1 ), (b02 , x0 , v 0 2 ) ∈ G(E2 ),
b01 + b02 = b0 , v 0 1 + v 0 2 = v 0 .
Then
hv − v 0 , x − x0 i = hv 1 − v 0 1 , x − x0 i + hv 2 − v 0 2 , x − x0 i
> −(b1 + b01 ) − (b2 + b02 )
= −(b + b0 ).
2
COROLLARY 7.3. Under the hypotheses of Lemma 7.2,
G(S T1 +T2 ) ⊆ G(S T1 ⊕ S T2 ) ⊆ G(B T1 ⊕ B T2 ) ⊆ G(B T1 +T2 ).
fam-enlarg.tex; 1/01/2000; 1:09; p.18
A Family of Enlargements
19
Acknowledgements
We are indebted to the anonymous referee for the constructive criticism
on the original version of this paper.
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