ε-Enlargements of maximal monotone operators in
Banach spaces
Regina Sandra Burachik ([email protected])∗
Engenharia de Sistemas e Computação, COPPE–UFRJ, CP 68511, Rio de
Janeiro–RJ, 21945–970, Brazil.
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. Given a maximal monotone operator T in a Banach space, we consider
an enlargement T ε , in which monotonicity is lost up to ε, in a very similar way to the
ε-subdifferential of a convex function. We establish in this general framework some
theoretical properties of T ε , like a transportation formula, local Lipschitz continuity,
local boundedness, and a Brøndsted & Rockafellar property.
Keywords: Banach spaces, maximal monotone operators, enlargement of an operator, Brøndsted & Rockafellar property, transportation formula, Lipschitz continuity,
local boundedness.
Mathematics Subject Classification (1991): 47H05, 46B99
1. Introduction and motivation
Let A and B be arbitrary sets and F : A → 2B a multifunction. By an
enlargement or extension of F we mean a multifunction E : R+ × A →
2B 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) = {u ∈ X ∗ : f (y) − f (x) − hu, y − xi ≥ 0 for all y ∈ X} .
The ε-subdifferential enlargement (of ∂f ) was introduced in [1]. It is
defined as
∂ε f (x) := {u ∈ X ∗ : f (y) − f (x) − hu, y − xi ≥ −ε for all y ∈ X} ,
∗
Partially supported by PRONEX–Optimization.
Partially supported by CNPq Grant 301200/93-9(RN) and by PRONEX–
Optimization.
†
c 1999 Kluwer Academic Publishers. Printed in the Netherlands.
teps-banach.tex; 1/01/1999; 22:11; p.1
2
for any ε ≥ 0, x ∈ X. Note that ∂f = ∂0 f and in general ( i.e., for
ε ≥ 0) ∂f (x) ⊆ ∂ε f (x). If f is proper closed convex, then ∂f is a
maximal monotone operator (see [12]). So, we have an example of an
enlargement of certain maximal monotone operators (those which are
subdifferentials).
N
For an arbitrary maximal monotone operator T : RN → 2(R ) ,
Burachik, Iusem and Svaiter [3] proposed the following enlargement:
Given ε ≥ 0 and x ∈ RN
T ε (x) = {u ∈ RN | hv − u, y − xi ≥ −ε, ∀y ∈ RN , v ∈ T (y)} .
The enlargement defined above can be formulated in a straightforward
manner for operators defined on Hilbert spaces. In the case of a Banach
space X, the scalar product is replaced by the usual “dual product” on
X × X ∗ . Hence, for an arbitrary maximal monotone operator T : X →
∗
2X , we consider
T ε (x) = {u ∈ X ∗ | hv − u, y − xi ≥ −ε, ∀y ∈ X, v ∈ T (y)},
for any ε ≥ 0, x ∈ X. The aim of this paper is to study the basic
properties of this enlargement for maximal monotone operators defined
in Banach spaces.
The idea of relaxing monotonicity has been explored in [13]. According to this work, an operator W is ε-monotone if
hu − v, x − yi ≥ −ε
∀ u ∈ W (x), v ∈ W (y).
Observe that in the above definition there is no maximal monotone operator to be extended. Furthermore, given T a maximal monotone operator, there are in general many maximal ε-monotone multifunctions
whose graph contain the graph of T .
An important property concerning any extension of a maximal monotone operator T is whether an element in the graph of the extension
of T can be approximated by an element in the graph of the original
operator. This question has been successfully solved for the extension
∂ε f by Brøndsted & Rockafellar in [1]: Given ε ≥ 0 and vε ∈ ∂ε f (xε ),
for any η > 0, there exists x and v ∈ ∂f (x) such that
kx − xε k ≤ ε/η,
kv − vε k ≤ η.
We say that an enlargement E of T satisfies the Brøndsted & Rockafellar property if any element in the graph of E can be approximated in a
similar way, i.e.: Given b ≥ 0, vb ∈ E(b, xb ), for any η > 0 there exists
x and v ∈ T (x) such that
kx − xb k ≤ ε/η,
kv − vb k ≤ η.
teps-banach.tex; 1/01/1999; 22:11; p.2
ε-Enlargements in Banach spaces
3
When X is a Hilbert space, it has been proved in [4], that the enlargement defined above satisfies the Brøndsted & Rockafellar property. In
this work, we prove that T ε has this property for any reflexive Banach
space.
Another key question concerning an extension of T is how to construct an element of the extension using elements of T . For the extension ∂ε f , the tool for doing this is the so-called “transportation
formula” (see Proposition 4.2.2, Vol.2 [8]). For the extension T ε , a
transportation formula was found in [4] for operators defined on Hilbert
spaces. In this work, we will show that this formula still holds for
operators defined on Banach spaces. This “alternative” transportation
formula was already obtained for the extension ∂ε f in [10] (Proposition 1.2.10).
Finally, we prove that (ε, x) ,→ T ε (x) is locally bounded by an affine
function of ε on the interior of the domain of T , as well as Lipschitz
continuous, for T defined in a Banach space. We point out that this
result was proved in [4] for X = RN .
Algorithmic applications of this enlargement can be found in [3], [4]
and [5].
The paper is organized as follows. In Section 2 we give the theoretical preliminaries. The main results are established in Section 3:
The Brøndsted & Rockafellar property, the “transportation formula”,
the “ε-affine” local boundedness and the Lipschitz-continuity of the
multifunction (ε, x) ,→ T ε (x).
2. Basic definitions
¿From now on X is a real Banach space, X ∗ is its dual. When X is
reflexive, X ∗∗ (the dual of X ∗ ) is identified with X. Given x ∈ X and
v ∈ X ∗ , v(x) will be denoted indifferently by
p hx, vi and hv, xi. The
norm in X × X ∗ will be taken as k(x, v)k = kxk2 + kvk2 . We need
first some notation.
Let R+ := {α ∈ R | α ≥ 0}. For ρ ≥ 0, we denote by B(x, ρ) the open
ball with center at x and radius ρ, i.e., B(x, ρ) := {x ∈ X : kxk < ρ}.
∗
Given a set A ⊆ X and a multifunction S : X → 2X ,
− we define the set S(A) :=
S
a∈A S(a) .
− The domain, image and graph of S are respectively denoted by
D(S) := {x ∈ X : S(x) 6= ∅} ,
R(S) := S(X) and
G(S) := {(x, v) : x ∈ X and v ∈ S(x)} .
teps-banach.tex; 1/01/1999; 22:11; p.3
4
− The inverse of S is the multifunction
S −1 : X ∗ → 2X
defined by
S −1 (v) = {x ∈ X : v ∈ S(x)} .
− S is locally bounded at x if there exists a neighborhood U of x such
that the set S(U ) is bounded.
− S is monotone if hu − v, x − yi ≥ 0 for all u ∈ S(x) and v ∈ S(y),
for all x, y ∈ X.
∗
− S: X → 2X is maximal monotone on X if it is monotone and its
graph is maximal with respect to this property, i.e., its graph is not
properly contained in the graph of any other monotone operator.
Based on [3], we define the enlargement of monotone operators on
a Banach space:
∗
DEFINITION 2.1. Let T : X → 2X be monotone. The ε-enlargement
of T is the multifunction
T (·) (·) : R+ × X → 2X
∗
defined by
T ε (x) = {u ∈ X ∗ : hv − u, y − xi ≥ −ε , ∀y ∈ X , v ∈ T (y)}
(1)
for x ∈ X and ε ≥ 0. We will also use the notation
∗
T ε (·) = T (ε, ·) : X → 2X .
Although some results below hold for the above defined enlargement
of arbitrary monotone multifunctions, we are concerned in this paper
with maximal monotone operators. Therefore, from now on,
T : X → 2X
∗
is a maximal monotone operator.
Observe that if T = ∂f , for some proper closed and convex function
f , then ∂ε f (x) ⊆ T ε (x). This result, as well as examples showing that
the inclusion can be strict, can be found in [3].
Note that ε is used to define different concepts and objects: εenlargement, ε-subdifferential, ε-monotonicity, ε solutions etc. Note
also that ε is also a real variable. To avoid confusion, from now on
we will try not to use ε as a variable.
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ε-Enlargements in Banach spaces
5
3. Properties of T ε
The following properties, which also can be found in [3], still hold in
Banach spaces.
LEMMA 3.1.
1. T 0 (·) = T (·)
2. For any b1 , b2 ∈ R+ , if b1 ≤ b2 then
T (b1 , x) ⊆ T (b2 , x),
∀x ∈ X.
3. If I ⊆ R+ is nonempty then
∩b∈I T (b, x) = T (b̄, x),
∀x ∈ X.
where b̄ = inf I.
4. T b (x) is weak∗closed and convex for any b ≥ 0, x ∈ X.
Proof. The proof follows directly from Definition 2.1. We point out
that only item 1 depends on the maximal monotonicity of T .
∗
Since T : X → 2X is maximal monotone, T −1 : X ∗ → 2X is
monotone. If X is reflexive then T −1 is maximal monotone. In this
case the ε-enlargements of T and T −1 are quite connected.
LEMMA 3.2. If X is a reflexive Banach space, then for any b ≥ 0
(T −1 )b = (T b )−1 .
Proof. The proof is straightforward from Definition 2.1.
Given a closed and convex function f and a positive α, it follows
easily from the definition of ε-subdifferential that ∂ε (αf ) = α ∂(ε/α) (f ).
This property can be easily established also for the multifunction T b .
More specifically:
LEMMA 3.3. For any α > 0,
(αT )b = α T b/α .
It is well-known (see [7], Proposition 1.3) that the following relation between the epsilon subdifferential of a sum of two proper convex
lower-semicontinuous functions f1 , f2 and the sum of the corresponding
teps-banach.tex; 1/01/1999; 22:11; p.5
6
epsilon subdifferentials holds: If x0 ∈ dom f1 ∩ dom f2 and one of the
functions is continuous at this point then, given ε > 0,
∂ε (f1 + f2 )(x0 ) = ∪θ∈[0,1] ∂θε f1 (x0 ) + ∂(1−θ)ε f2 (x0 ) .
For the T ε enlargement, however, we recover only one of the inclusions.
Suppose T1 , T2 and T1 + T2 maximal monotone. Direct application of
Definition 2.1 yields: for any x ∈ D(T1 ) ∩ D(T2 ) and b ≥ 0,
(1−θ)b
(T1 + T2 )b (x) ⊃ ∪θ∈[0,1] T1θb (x) + T2
(x) .
It is well-known that the graph of a maximal monotone operator is
demiclosed. This property is shared by the graph of T (·, ·).
∗
PROPOSITION 3.4. The graph of T (·, ·) : R+ × X → 2X is demiclosed, i.e., the conditions below hold.
(a) If {xk } ⊂ X converges strongly to x0 , {uk ∈ T (bk , xk )} converges weak ∗ to u0 in X ∗ and {bk } ⊂ R+ converges to b, then
u0 ∈ T (b, x0 ).
(b) If {xk } ⊂ X converges weakly to x0 , {uk ∈ T (bk , xk )} converges
strongly to u0 in X ∗ and {bk } ⊂ R+ converges to b, then u0 ∈
T (b, x0 ).
Proof.
To prove (a), observe that since {xk } converges strongly and {uk }
converges weak∗, they are bounded sequences. So, let M < +∞ be a
bound for kxk k and kuk k. Take any (y, v) ∈ G(T ). From Definition 2.1,
it follows that
−bk ≤
=
=
≤
hxk − y, uk − vi
hxk − y, u0 − vi + hxk − y, uk − u0 i
hxk − y, u0 − vi + hxk − x0 , uk − u0 i + hx0 − y, uk − u0 i
hxk − y, u0 − vi + kxk − x0 k(M + ku0 k) + hx0 − y, uk − u0 i
Taking the limit k → ∞, we obtain
−b ≤ hx0 − y, u0 − vi.
Since (y, v) is an arbitrary element in the graph of T , the conclusion
follows.
For proving item (b), recall that weak convergent sequences are also
bounded and use a similar reasoning.
teps-banach.tex; 1/01/1999; 22:11; p.6
ε-Enlargements in Banach spaces
7
3.1. Brøndsted & Rockafellar Property
For a closed proper convex function f , the theorem of Brøndsted &
Rockafellar (see [1]), states that any ε-subgradient of f at a point xε
can be approximated by some exact subgradient, computed at some
x, possibly different from xε .
The T ε enlargement also satisfies this property. This fact has been
proved for a Hilbert space in [4]. We extend below this result to a
reflexive Banach space.
DEFINITION 3.5. Consider g : X → R defined by
1
g(x) = kxk2 .
2
∗
The duality mapping J : X → 2X is defined by J(x) := ∂g(x).
As a subdifferential of a convex function, J is maximal monotone. We
recall below two important results concerning the duality mapping J.
PROPOSITION 3.6.
(i) J(x) = {v ∈ X ∗ | hx, vi = kxk kvk, kxk = kvk},
(ii) Let X be reflexive, and take S any maximal monotone operator
on X. Then S + λJ is onto for any λ > 0.
The proof of (i) can be found in [6]. Item (ii) follows from a result of
Browder [2], Theorem 7.2.
THEOREM 3.7. Assume X is a reflexive Banach space. Let b ≥ 0 and
(xb , vb ) ∈ G(T b ). Then for all η > 0 there exists (x, v) ∈ G(T ) such
that
kv − vb k ≤
b
η
and
kx − xb k ≤ η .
(2)
Proof. If b = 0, then (2) holds with (x, v) = (xb , vb ) ∈ G(T ). Suppose
now b > 0. For an arbitrary positive coefficient β define the operator
∗
G β : X → 2X
y 7→ βT (y) + {J(y − xb )} ,
where J is the duality operator of X. Since βT is maximal monotone,
by Proposition 3.6(ii), Gβ is onto. In particular βvb is in the image of
this operator and there exist x ∈ X and v ∈ T (x) such that
βvb ∈ βv + J(x − xb ) .
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8
Therefore β(vb − v) ∈ J(x − xb ). This, together with Proposition 3.6(i)
and Definition 2.1, yields
1
hvb − v, xb − xi = − kx − xb k2
β
= −βkv − vb k2 ≥ −b .
Choosing β := η 2 /b, the result follows.
Observe that the proof above only uses the inequality which characterizes elements in T b . This means that the Brøndsted & Rockafellar
property holds for any extension ET of T which satisfies ET (b, ·) ⊂
T b (·). As ∂b f (·) ⊂ (∂f )b (·), we recover Brøndsted & Rockafellar’s theorem for b-subdifferentials in a reflexive Banach space.
The following corollary, which extends slightly Proposition 2 in [3],
establishes a relation between the image, domain and graph of an
operator and its extension T ε .
COROLLARY 3.8. Let X be a reflexive Banach space. The following
inclusions hold.
(i) R(T ) ⊂ R(T b ) ⊂ R(T ),
(ii) D(T ) ⊂ D(T b ) ⊂ D(T ),
(iii) If d(· ; ·) denotes the point-to-set distance, then
√
d((xb , vb ); G(T )) ≤ 2b ,
whenever (xb , vb ) ∈ G(T b ).
Proof. The leftmost inclusions in (i) and (ii) are straightforward
from Definition 2.1. As for the right ones, they follow from Theorem 3.7,
making η → +∞ and η → 0√in (i) and (ii) respectively.
To prove (iii), take η = b in (2), write
d((xb , vb ); G(T ))2 ≤ kx − xb k2 + kv − vb k2 ≤ 2b ,
and take square roots.
3.2.
“ε-affine” local boundedness
It is a well-known fact (see [9],[11]) that any maximal monotone operator in an arbitrary Banach space is locally bounded on the interior
of its domain. The following result establishes that also the extended
operator T ε is locally bounded on the interior of the domain of T , by
an affine function of ε. In order to prove this, we start with a technical
result.
teps-banach.tex; 1/01/1999; 22:11; p.8
9
ε-Enlargements in Banach spaces
PROPOSITION 3.9. Let U ⊂ D(T ). Define
M := sup{kuk | u ∈ T (U )}.
(3)
If V ⊂ X and ρ > 0 are such that
V + B(0, ρ) ⊂ U,
then
sup{kvk | v ∈ T (b, V )} ≤
(4)
b
+ M,
ρ
(5)
for any b ≥ 0.
Proof. Observe that (4) is equivalent to the following inclusion:
{x ∈ X | d(x, V ) < ρ} ⊂ U.
(6)
Take x̃ ∈ V and ṽ ∈ T (b, x̃). Recall that the norm in X ∗ is given by
kṽk := sup{hz, ṽi | z ∈ X , kzk = 1}.
Hence there exists a sequence {z k } ⊂ X such that kz k k = 1 and
lim hz k , ṽi = kṽk.
k→∞
Define y k := x̃ + σz k , where σ ∈ (0, ρ). Clearly
d(y k , V ) ≤ ky k − x̃k = σ < ρ.
Hence, by our assumption, {y k } ⊂ U . Using this fact and the definition
of M , we conclude that for any wk ∈ T (y k ), kwk k ≤ M . By definition
of T b and y k ,
−b ≤ hṽ − wk , x̃ − y k i = hṽ − wk , −σz k i.
Dividing the expression above by σ, we get
−
b
≤ −hṽ, z k i + hwk , z k i ≤
σ
−hṽ, z k i + kwk kkz k k =
−hṽ, z k i + kwk k ≤ −hṽ, z k i + M,
(7)
where we used the definitions of {z k } and M .
Rearranging (7), we obtain
hṽ, z k i ≤
b
+ M.
σ
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10
Taking now limit for k tending to infinity, we obtain a bound for ṽ:
kṽk ≤
b
+ M.
σ
Since the inequality above is true for any σ < ρ, we conclude that
kṽk ≤
b
+ M.
ρ
Now we are able to prove a property of T ε that is slightly stronger
than local boundedness. This property will be useful in the sequel (for
proving Lipschitz continuity).
COROLLARY 3.10 (“ε-affine” local boundedness). For any x in the
interior of D(T ) there exist a neighborhood of x, V ⊂ int D(T ) and
constants L, M > 0, such that
sup{kvk | v ∈ T ε (V )} ≤ εL + M
for any ε ≥ 0.
Proof. Take x ∈ int D(T ). By [11], Theorem 1, T is locally bounded
on int D(T ). Hence, for some R > 0, B(x, R) ⊆ D(T ) and T is bounded
on B(x, R). Take
ρ := R/2; V := B(x, ρ)
and apply Proposition 3.9.
3.3. Transportation Formula
We already mentioned that the set T (b, x) approximates T (x), but this
fact is of no use as long as there is no way of computing elements of
T (b, x). The question is then how to construct an element in T (b, x)
with the help of some elements (xi , v i ) ∈ G(T ). The answer is given by
the “transportation formula” stated below. Therein we use the notation
∆m := {α ∈ Rm | αi ≥ 0 ,
m
X
αi = 1}
i=1
for the unit-simplex in Rm .
THEOREM 3.11. Consider a set of m triplets
{(bi , xi , v i ∈ T (bi , xi ))}i=1,...,m .
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11
ε-Enlargements in Banach spaces
For any α ∈ ∆m define
i
x̂ := Pm
1 αi x
m
i
v̂ :=
1 αi v
Pm
Pm
i
i
b̂ :=
1 αi bi +
1 αi hx − x̂, v − v̂i .
P
(8)
Then b̂ ≥ 0 and v̂ ∈ T (b̂, x̂).
Proof. Recalling Definition 2.1, we need to show that
hx̂ − y, v̂ − vi ≥ −b̂ ,
for any (y, v) ∈ G(T ). Combine (8) and (1), with (b, x, u) replaced by
(bi , xi , v i ), to obtain
i
hx̂ − y, v̂ − vi = Pm
1 αi hx − y, v̂ − vi
m
hxi − y, v i − vi]
= P1 αi [hxi − y, v̂ − v i i +P
m
m
i
i
≥
1 αi bi .
1 αi hx − y, v̂ − v i −
P
(9)
Since
Pm
1
m
i
i
i
αi hxi − y, v̂ − v i i = P
1 αi [hx − x̂, v̂ − v i + hx̂ − y, v̂ − v i]
m
i
i
= − P1 αi hx − x̂, v − v̂i + 0
i
i
= − m
1 αi hx − x̂, v − v̂i ,
P
with (9) and (8) we get
hx̂ − y, v̂ − vi ≥ −b̂ .
(10)
For contradiction purposes, suppose that b̂ < 0. Then hx̂−y, v̂ −vi >
0 for any (y, v) ∈ G(T ) and the maximality of T implies that (x̂, v̂) ∈
G(T ). In particular, the pair (y, v) = (x̂, v̂) yields 0 > 0. Therefore
b̂ must be nonnegative. Since (10) holds for any (y, v) ∈ G(T ), we
conclude from (1) that v̂ ∈ T (b̂, x̂).
Observe that when bi = 0, for all i = 1, . . . , m, this theorem shows
how to construct v̂ ∈ T (b̂, x̂), using (xi , v i ) ∈ G(T ).
The formula above holds also when replacing T b̂ by ∂b̂ f , with f a
proper closed and convex function. This is Proposition 1.2.10 in [10],
where an equivalent expression is given for b̂:
m
X
m
1 X
b̂ =
αi bi +
αi αj hxi − xj , v i − v j i .
2
i=1
i,j=1
Observe also that, when compared to the standard transportation
formula for ε-subdifferentials, Theorem 3.11 is a weak transportation
teps-banach.tex; 1/01/1999; 22:11; p.11
12
formula, in the sense that it only allows us to express some selected
ε-subgradients in terms of subgradients.
The transportation formula can also be used for the ε-subdifferential
to obtain the lower bound:
If {(v i ∈ ∂bi f (xi ))}i=1,2
then hx1 − x2 , v 1 − v 2 i ≥ −(b1 + b2 ) .
For the enlargement T ε , we have a weaker bound:
COROLLARY 3.12. Take v 1 ∈ T (b1 , x1 ) and v 2 ∈ T (b2 , x2 ). Then
hx1 − x2 , v 1 − v 2 i ≥ −( b1 +
p
b2 )2 .
p
(11)
Proof. If b1 or b2 are zero the result holds trivially. Otherwise choose
α ∈ ∆2 as follows
√
√
b2
b1
√
√
α1 := √
α2 := 1 − α1 = √
(12)
b1 + b 2
b1 + b2
and define the convex sums x̂, v̂ and b̂ as in (8). Because b̂ ≥ 0, we can
write
0 ≤ b̂ = α1 b1 + α2 b2 + α1 hx1 − x̂, v 1 − v̂i + α2 hx2 − x̂, v 2 − v̂i
= α1 b1 + α2 b2 + α1 α2 hx2 − x1 , v 2 − v 1 i ,
where we have used the identities x1 − x̂ = α2 (x1 − x2 ), x2 − x̂ =
α1 (x2 − x1 ), v 1 − v̂ = α2 (v 1 − v 2 ) and v 2 − v̂ = α1 (v 2 − v 1 ) first, and
then α1 α22 + α12 α2 = α1 α2 . Now, combine the expression above with (8)
and (12) to obtain
√
p
b1 b2
√
b1 b 2 + √
hx1 − x2 , v 1 − v 2 i ≥ 0 .
( b1 + b2 )2
Rearranging terms and simplifying the resulting expression, (11) is
proved.
Observe that the result above implies that T ε is 4ε-monotone in the
sense of [13].
3.4. Lipschitz Continuity
For the sake of generality, we will work with any ET , extension of T ,
which satisfies:
(E1 ) For any (b, x) ∈ R+ × X , T (x) ⊂ ET (b, x) ⊂ T (b, x),
(E2 ) If 0 ≤ b1 ≤ b2 , then ET (b1 , x) ⊂ ET (b2 , x) for any x ∈ X.
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ε-Enlargements in Banach spaces
13
(E3 ) The transportation formula holds for ET (·, ·). More precisely, let
{(bi , xi , v i ∈ ET (bi , xi ))}i=1,...,m ,
be any set of m triplets on the graph of ET (·, ·) and take α ∈ ∆m .
Consider
P
i
x̂ := Pm
1 αi x
m
i
v̂ :=
(13)
1 αi v
Pm
Pm
i
i
b̂ :=
1 αi bi +
1 αi hx − x̂, v − v̂i .
Then b̂ ≥ 0 and v̂ ∈ ET (b̂, x̂).
Our aim is to prove that any such extension is Lipschitz continuous.
REMARK 3.13. Observe that for T = ∂f , the extension
ET (b, x) = ∂b f (x)
satisfies (E1 ) − (E3 ). This fact will allow us to give an alternative proof
of the Lipschitz continuity of ∂b f (x) through Theorem 3.14.
For an arbitrary maximal monotone operator T , the extension
ET (b, x) = T b (x)
satisfies (E1 ) − (E3 ) by definition of T ε , Lemma 3.1 and Theorem 3.11.
For the extension ∂ε f (·) the Lipschitz continuity was proved in [7].
For the extension T ε in a finite dimensional space, this property was
established in [4].
We recall that a closed-valued locally bounded multifunction S is
continuous at x̄ if for any positive there exists δ > 0 such that
kx − x̄k ≤ δ
=⇒
S(x) ⊂ S(x̄) + B(0, ) ,
S(x̄) ⊂ S(x) + B(0, ) .
Furthermore, S is Lipschitz continuous on V if there exists a nonnegative constant L such that for any y 1 , y 2 ∈ V and s1 ∈ S(y 1 ) there exists
s2 ∈ S(y 2 ) satisfying ks1 − s2 k ≤ Lky 1 − y 2 k.
For simplicity of notation, we write D for the interior of D(T ).
THEOREM 3.14. Let ET be an extension of T which satisfies (E1 ) −
(E3 ). Take V ⊂ X and ρ > 0. Suppose that T is bounded on U ⊂ D
and that
{x ∈ X | d(x, V ) < ρ} ⊂ U.
(14)
Then ET is Lipschitz continuous on [b, b]×V . In other words, if b, b ≥ 0
are such that 0 < b ≤ b < +∞, then there exist nonnegative constants
A and B such that for any (b1 , x1 ), (b2 , x2 ) ∈ [b, b] × V and v 1 ∈
ET (b1 , x1 ), there exists v 2 ∈ ET (b2 , x2 ) satisfying
kv 1 − v 2 k ≤ Akx1 − x2 k + B|b1 − b2 | .
(15)
teps-banach.tex; 1/01/1999; 22:11; p.13
14
Proof. Define
M := sup{kuk | u ∈ T (U )},
(16)
which is finite by assumption. We claim that (15) holds for the following
choice of A and B:
A :=
1 2M
+
ρ
b
!
b
+ 2M ,
ρ
B :=
1 2M
+
ρ
b
.
(17)
To see this, take x1 , x2 , b1 , b2 and v 1 as above. Take l := kx1 − x2 k and
let x3 be in the line containing x1 and x2 such that
kx3 − x2 k = ρ ,
kx3 − x1 k = ρ + l ,
(18)
as shown in Figure 1.
r
l
-
r
x1
ρ
r
x2
x3
Figure 1.
Then d(x3 , V ) ≤ d(x3 , x2 ) = ρ and hence x3 ∈ U by (14). By
definition of x3 , it is straightforward that
x2 = (1 − θ)x1 + θx3 ,
with
θ=
l
∈ [0, 1) .
ρ+l
Now, take u3 ∈ T (x3 ) and define
ṽ 2 := (1 − θ)v 1 + θu3 .
By property (E3 ), ṽ 2 ∈ ET (b̃2 , x2 ), with
b̃2 = (1 − θ)b1 + (1 − θ)hx
1 − x2 , v 1 − ṽ 2 i +
θ x3 − x2 , u3 − ṽ 2
= (1 − θ)b1 + θ(1 − θ) x1 − x3 , v 1 − u3 .
Use Proposition 3.9 with v 1 ∈ ET (b1 , x1 ) ⊂ ET (b1 , V ) ⊂ T (b1 , V ),
together with the definition of M , to obtain
kv 1 − u3 k ≤ kv 1 k + ku3 k ≤ (
b1
b1
+ M) + M ≤
+ 2M ,
ρ
ρ
(19)
where we are using also that u3 ∈ T (U ). Using that the dual product
on X × X ∗ satisfies hx, vi ≤ kxkkvk, (19), (18), and the definition of θ,
we get
3k
b̃2 ≤ (1 − θ)b1 + θ(1 − θ)kx1 − x3 k kv 1 − u
b1
≤ (1 − θ)b1 + θ(1 − θ)(ρ + l)
+ 2M
(20)
ρ
ρl
= b1 +
2M.
ρ+l
teps-banach.tex; 1/01/1999; 22:11; p.14
15
ε-Enlargements in Banach spaces
The definition of ṽ 2 combined with (19) yields
kv 1
−
ṽ 2 k
b1
=
−
≤θ
+ 2M
ρ
1
2 1 b1
+ 2M ,
≤ kx − x k
ρ ρ
θkv 1
u3 k
(21)
where we used the definition of θ in the last inequality. Now consider
two cases:
(i) b̃2 ≤ b2 ,
(ii) b̃2 > b2 .
If (i) holds, by (E2 ), ṽ 2 ∈ ET (b̃2 , x2 ) ⊆ ET (b2 , x2 ). Then, choosing
:= ṽ 2 and using (21) together with (17), (15) follows.
b2
< 1 and v 2 := (1 − β)u2 + βṽ 2 , with
In case (ii), define β :=
b̃2
u2 ∈ T (x2 ). Because of (E3 ), v 2 ∈ ET (b2 , x2 ) ⊂ ET (b2 , V ). On the
other hand, as u2 ∈ T (x2 ), and x2 ∈ V ⊂ U , we obtain, in a similar
way as in (19):
b1
+ 2M .
(22)
ku2 − v 1 k ≤
ρ
v2
Combining the definition of v 2 and (22), we obtain
kv 2 − v 1 k ≤ (1 − β)ku2 − v 1 k + βkṽ 2 − v 1 k
b1
b1
≤ (1 − β)
+ 2M + βθ
+ 2M
ρ
ρ
b1
= (1 − β(1 − θ))
+ 2M .
ρ
(23)
Using (20) we have that
β=
b2
b2
≥
.
ρl
b̃2
b1 +
2M
ρ+l
Some elementary algebra, the inequality above and the definitions of θ
and l, yield
ρ(b1 − b2 )
b1 + ρ2M
+
(ρ + l)b1 + ρl2M
(ρ + l)b1 + ρl2M
1 2M
1
≤ kx1 − x2 k
+
+ |b1 − b2 | .
ρ
b1
b1
1 − β(1 − θ) ≤ l
(24)
Altogether, with (23), (24) and our assumptions on b1 , b2 , b, b, the conclusion follows.
teps-banach.tex; 1/01/1999; 22:11; p.15
16
As a result from Theorem 3.14, we obtain Lipschitz continuity on any
compact set contained in the interior of the domain of T .
COROLLARY 3.15. Let ET be an extension of T which satisfies (E1 )−
(E3 ). Let K ⊂ D be a compact set. Take b, b ≥ 0 such that 0 < b ≤
b < +∞. Then there exists an open set V ,
K⊂V ⊂D
such that ET is Lipschitz continuous on [b, b̄] × V , i.e., there exist
nonnegative constants A and B such that for any (b1 , x1 ), (b2 , x2 ) ∈
[b, b] × V and v 1 ∈ ET (b1 , x1 ), there exists v 2 ∈ ET (b2 , x2 ) satisfying
kv 1 − v 2 k ≤ Akx1 − x2 k + B|b1 − b2 | .
Proof. By a compactness argument, the local boundedness of T
implies that there exists an open set U such that K ⊂ U ⊂ D and
T (U ) is bounded. Using again the compactness of K it follows that for
some R > 0,
{x ∈ X | d(x, K) < R} ⊂ U.
(25)
Take
ρ = R/2 ,
V = {x ∈ X | d(x, K) < ρ} .
Then, K ⊂ V ⊂ D. Furthermore,
{x ∈ X | d(x, V ) < ρ} ⊂ U
and we can apply Theorem 3.14.
Acknowledgements
We are indebted to the anonymous referees for the corrections they
made on the original version of this paper.
References
1.
2.
Brøndsted, A. and Rockafellar, R.T.: On the subdifferentiability of convex
functions, Proceedings of the American Mathematical Society 16 (1965), 605–
611.
Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in
Banach spaces, Proceedings of Symposia in Pure Mathematics. 18(2) (1968),
American Mathematical Society.
teps-banach.tex; 1/01/1999; 22:11; p.16
ε-Enlargements in Banach spaces
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
17
Burachik, R.S., Iusem, A.N. and Svaiter, B.F.: Enlargements of maximal
monotone operators with application to variational inequalities, Set Valued
Analysis 5 (1997), 159–180.
Burachik, R.S., Sagastizábal, C.A. and Svaiter, B.F.: epsilon-Enlargement of
Maximal Monotone Operators with Application to Variational Inequalities, in
Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing
Methods. Fukushima, M. and Qi, L. (editors), Kluwer Academic Publishers,
1997.
Burachik, R.S., Sagastizábal, C.A. and Svaiter, B.F.: Bundle methods for maximal monotone operators, in Ill–posed Variational Problems and Regularization
Techniques. Tichatschke, R. and Thera, M. (editors), LNCIS Springer, 1999
(to appear).
Ciorănescu, I.: Aplicatii de dualitate in analiza functională neliniară. Ed.
Acad. Bucuresti, 1974.
Hiriart-Urruty, J.-B.: Lipschitz r-continuity of the approximate subdifferential
of a convex function, Math. Scand. 47 (1980), 123–134.
Hiriart-Urruty, J.-B. and Lemaréchal, C.: Convex Analysis and Minimization
Algorithms. Number 305-306 in Grund. der math. Wiss. Springer-Verlag, 1993.
(two volumes).
Kato, T.: Demicontinuity, hemicontinuity and monotonicity, Bull. Amer. Math.
Soc. 70 (1964), 548–550.
Lemaréchal, C.: Extensions diverses des méthodes de gradient et applications.
Thèse d’Etat, Université de Paris IX, 1980.
Rockafellar, R.T.: Local boundedness of nonlinear monotone operators,
Michigan Mathematical Journal 16 (1969), 397–407.
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings,
Pacific Journal Math. 33 (1970), 209–216.
Veselý, L.: Local uniform boundedness principle for families of ε-monotone
operators, Nonlinear Analysis, Theory, Methods & Applications 24(9) (1995),
1299–1304.
teps-banach.tex; 1/01/1999; 22:11; p.17
teps-banach.tex; 1/01/1999; 22:11; p.18