On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function

0.14 MB·English

by  Heinz H. Bauschke

On the maximal monotonicity of the sum
of a maximal monotone linear relation
and the subdifferential operator
0
1
0 of a sublinear function
2

n
a
J
Heinz H. Bauschke∗, Xianfu Wang†, and Liangjin Yao‡
1

]
A
January 1, 2010
F
.
h
t
a
m
[
Abstract

1
v
7 ThemostimportantopenprobleminMonotoneOperatorTheoryconcernsthemaximal
5 monotonicity of the sum of two maximal monotone operators provided that Rockafel-
2
lar’s constraint qualification holds.
0
. Inthisnote,weprovideanewmaximalmonotonicityresultforthesumofamaximal
1
0 monotone relation and the subdifferential operator of a proper, lower semicontinuous,
0 sublinear function. The proof relies on Rockafellar’s formula for the Fenchel conjugate
1
of the sum as well as some results on the Fitzpatrick function.
:
v
i
X
2000 Mathematics Subject Classification:
r
a
Primary 47A06, 47H05; Secondary 47B65, 49N15, 52A41, 90C25
Keywords: Constraint qualification, convex function, convex set, Fenchel conjugate, Fitz-
patrick function, linear relation, maximal monotone operator, multifunction, monotone op-
erator, set-valued operator, subdifferential operator, sublinear function, Rockafellar’s sum
theorem.
∗Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada.
E-mail: [email protected].
†Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada.
E-mail: [email protected].
‡Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada.
E-mail: [email protected].
1
1 Introduction
Throughout this paper, we assume that X is a real Banach space with norm k·k, that X∗
is the continuous dual of X, and that X and X∗ are paired by h·,·i. Let A: X ⇉ X∗ be
a set-valued operator (also known as multifunction) from X to X∗, i.e., for every x ∈ X,
Ax ⊆ X∗, and let graA = (x,x∗) ∈ X ×X∗ | x∗ ∈ Ax be the graph of A. Recall that A
is monotone if
(cid:8) (cid:9)
(1) ∀(x,x∗) ∈ graA ∀(y,y∗) ∈ graA hx−y,x∗ −y∗i ≥ 0,
and maximal mo(cid:0)notone if A is m(cid:1)(cid:0)onotone and A(cid:1)has no proper monotone extension (in
the sense of graph inclusion). We say A is a linear relation if graA is a linear subspace.
Monotone operators have proven to be a key class of objects in modern Optimization and
Analysis; see, e.g., the books [8, 9, 10, 13, 18, 19, 17, 26] and the references therein. (We also
adopt standard notation used in these books: domA = x ∈ X | Ax 6= ∅ is the domain
of A. Given a subset C of X, intC is the interior of C, and C is the closure of C. We set
(cid:8) (cid:9)
C⊥ := {x∗ ∈ X∗ | (∀c ∈ C)hx∗,ci = 0} and S⊥ := {x∗∗ ∈ X∗∗ | (∀s ∈ S)hx∗∗,si = 0} for a
set S ⊆ X∗. The indicator function of C, written as ι , is defined at x ∈ X by
C
0, if x ∈ C;
(2) ι (x) :=
C
(∞, otherwise.
Given f: X → ]−∞,+∞], we set domf = f−1(R) and f∗: X∗ → [−∞,+∞] : x∗ 7→
sup (hx,x∗i − f(x)) is the Fenchel conjugate of f. If f is convex and domf 6= ∅, then
x∈X
∂f: X ⇉ X∗: x 7→ x∗ ∈ X∗ | (∀y ∈ X) hy −x,x∗i+f(x) ≤ f(y) is the subdifferential
operator of f. Recall that f is sublinear if f(0) = 0, f(x+y) ≤ f(x)+f(y), and f(λx) =
(cid:8) (cid:9)
λf(x) for all x,y ∈ domf and λ > 0. Finally, the closed unit ball in X is denoted by
B := x ∈ X | kxk ≤ 1 .) Throughout, we shall identify X with its canonical image in
X
the bidual space X∗∗. Furthermore, X ×X∗ and (X ×X∗)∗ = X∗×X∗∗ are likewise paired
(cid:8) (cid:9)
via h(x,x∗),(y∗,y∗∗)i = hx,y∗i+hx∗,y∗∗i, where (x,x∗) ∈ X ×X∗ and (y∗,y∗∗) ∈ X∗×X∗∗.
LetAandB bemaximalmonotoneoperatorsfromX toX∗. Clearly, thesum operator A+
B: X ⇉ X∗: x 7→ Ax+Bx = a∗ +b∗ | a∗ ∈ Ax and b∗ ∈ Bx is monotone. Rockafellar’s
[16, Theorem 1] guarantees maximal monotonicity of A + B under the classical constraint
(cid:8) (cid:9)
qualification domA∩intdom B 6= ∅ when X is reflexive. The most famous open problem
concerns the behaviour in nonreflexive Banach spaces. See Simons’ monograph [19] for a
comprehensive account of the recent developments.
Now we focus on the special case when A is a linear relation and B is the subdifferential
operator of a sublinear function f. We show that the sum theorem is true in this setting.
Recently, linear relations have increasingly been studied in detail; see, e.g., [1, 2, 3, 4, 5, 6,
7, 14, 21, 23, 24, 25] and Cross’ book [11] for general background on linear relations.
2
The remainder of this paper is organized as follows. In Section 2, we collect auxiliary
results for future reference and for the reader’s convenience. The main result (Theorem 3.1)
is proved in Section 3.
2 Auxiliary Results
Fact 2.1 (Rockafellar) (See [15, Theorem 3], [19, Corollary 10.3 and Theorem 18.1], or
[26, Theorem 2.8.7(iii)].)
Let f,g : X → ]−∞,+∞] be proper convex functions. Assume that there exists a point
x ∈ domf ∩ domg such that g is continuous at x . Then for every z∗ ∈ X∗, there exists
0 0
y∗ ∈ X∗ such that
(3) (f +g)∗(z∗) = f∗(y∗)+g∗(z∗ −y∗).
Furthermore, ∂(f +g) = ∂f +∂g.
Fact 2.2 (Fitzpatrick) (See [12, Corollary 3.9].) Let A: X ⇉ X∗ be maximal monotone,
and set
(4) F : X ×X∗ → ]−∞,+∞] : (x,x∗) 7→ sup hx,a∗i+ha,x∗i−ha,a∗i ,
A
(a,a∗)∈graA
(cid:0) (cid:1)
which is the Fitzpatrick function associated with A. Then for every (x,x∗) ∈ X ×X∗, the
inequality hx,x∗i ≤ F (x,x∗) is true, and equality holds if and only if (x,x∗) ∈ graA.
A
Fact 2.3 (Simons) (See [19, Theorem 24.1(c)].) Let A,B : X ⇉ X∗ be maximal monotone
operators. Assume λ[P (domF )−P (domF )] is a closed subspace, where P :
λ>0 X A X B X
(x,x∗) ∈ X ×X∗ → x. If
S
(5) (x,x∗)is monotonically related to gra(A+B) ⇒ x ∈ domA∩domB,
then A+B is maximal monotone.
Fact 2.4 (Simons) (See[19, Lemma19.7andSection22].) Let A : X ⇉ X∗ be a monotone
linear relation such that graA 6= ∅. Then the function
(6) g: X ×X∗ → ]−∞,+∞] : (x,x∗) 7→ hx,x∗i+ι (x,x∗)
graA
is proper and convex.
3
Fact 2.5 (Simons) (See [20, Lemma 2.2].) Let f : X → ]−∞,+∞] be proper, lower
semicontinuous, and convex. Let x ∈ X and λ ∈ R be such that inff < λ < f(x) ≤ +∞,
and set
λ−f(a)
K := sup .
kx−ak
a∈X,a6=x
Then K ∈ ]0,+∞[ and for every ε ∈ ]0,1[, there exists (y,y∗) ∈ gra∂f such that
(7) hy −x,y∗i ≤ −(1−ε)Kky −xk < 0.
Fact 2.6 (See [26, Therorem 2.4.14].) Let f : X → ]−∞,+∞] be a sublinear function.
Then the following hold.
(i) ∂f(x) = {x∗ ∈ ∂f(0) | hx∗,xi = f(x)}, ∀x ∈ domf.
(ii) ∂f(0) 6= ∅ ⇔ f is lower semicontinuous at 0.
(iii) If f is lower semicontinuous, then f = suph·,∂f(0)i.
Fact 2.7 (See [13, Proposition 3.3 and Proposition 1.11].) Let f : X → ]−∞,+∞] be a
lower semicontinuous convex and intdom f 6= ∅. Then f is continuous on intdom f and
∂f(x) 6= ∅ for every x ∈ intdom f.
Lemma 2.8 Let f : X → ]−∞,+∞] be a sublinear function. Then domf +intdom f =
intdom f.
Proof. The result is trivial when intdom f = ∅ so we assume that x ∈ intdom f. Then
0
there exists δ > 0 such that x + δB ⊆ domf. By sublinearity, ∀y ∈ domf, we have
0 X
y +x +δB ⊆ domf. Hence
0 X
y +x ∈ intdomf.
0
Then domf + intdom f ⊆ intdom f. Since 0 ∈ domf, intdom f ⊆ domf + intdom f.
Hence domf +intdom f = intdom f. (cid:4)
Lemma 2.9 Let A : X ⇉ X∗ be a maximal monotone linear relation, and let z ∈ X∩(A0)⊥.
Then z ∈ domA.
Proof. Suppose to the contrary that z ∈/ domA. Then the Separation Theorem provides
w∗ ∈ X∗ such that
(8) hz,w∗i > 0 and w∗ ∈ domA⊥.
4
Thus, (0,w∗) is monotonically related to graA. Since A is maximal monotone, we deduce
that w∗ ∈ A0. By assumption, hz,w∗i = 0, which contradicts (8). Hence, z ∈ domA. (cid:4)
The proof of the next result follows closely the proof of [19, Theorem 53.1].
Lemma 2.10 Let A : X ⇉ X∗ be a monotone linear relation, and let f : X → ]−∞,+∞]
be a proper lower semicontinuous convex function. Suppose that domA ∩ intdom∂f 6= ∅,
(z,z∗) ∈ X × X∗ is monotonically related to gra(A + ∂f), and that z ∈ domA. Then
z ∈ dom∂f.
Proof. Let c ∈ X and y∗ ∈ X∗ be such that
0
(9) c ∈ domA∩intdom∂f and (z,y∗) ∈ graA.
0
Take c∗ ∈ Ac , and set
0 0
(10) M := max ky∗k,kc∗k ,
0
D := [c ,z], and h := f + ι . By (9), F(cid:8)act 2.7 an(cid:9)d Fact 2.1, ∂h = ∂f + ∂ι . Set
0 D D
H: X → ]−∞,+∞] : x 7→ h(x+z)−hz∗,xi. It remains to show that
(11) 0 ∈ dom∂H.
If infH = H(0), then (11) holds. Now suppose that infH < H(0). Let λ ∈ R be such that
infH < λ < H(0), and set
λ−H(x)
(12) K := sup .
λ
kxk
H(x)<λ
We claim that
K ≤ M.
λ
By Fact 2.5, we have K ∈ ]0,∞[ and ∀ε ∈ ]0,1[, by gra∂H = gra∂h−(z,z∗) there exists
λ
(x,x∗) ∈ gra∂h such that
(13) hx−z,x∗ −z∗i ≤ −(1−ε)K kx−zk < 0.
λ
Since ∂h = ∂f + ∂ι , there exists t ∈ [0,1] with x∗ ∈ ∂f(x) and x∗ ∈ ∂ι (x) such that
D 1 2 D
x = tc +(1−t)z and x∗ = x∗ +x∗. Then hx−z,x∗i ≥ 0. Thus, by (13),
0 1 2 2
(14) hx−z,x∗ −z∗i ≤ hx−z,x∗ +x∗ −z∗i ≤ −(1−ε)K kx−zk < 0.
1 1 2 λ
As x = tc +(1−t)z and A is a linear relation, we have (x,tc∗ +(1−t)y∗) ∈ graA. Since
0 0
(z,z∗) is monotonically related to gra(A+∂f), by (10),
(15) hx−z,x∗ −z∗i ≥ −hx−z,tc∗ +(1−t)y∗i ≥ −Mkx−zk.
1 0
5
Combining (15) and (14), we obtain
(16) −Mkx−zk ≤ −(1−ε)K kx−zk < 0.
λ
Hence, (1−ε)K ≤ M. Letting ε ↓ 0, we deduce that K ≤ M. Then, by (12) and letting
λ λ
λ ↑ H(0), we get
(17) H(y)+Mkyk ≥ H(0), ∀y ∈ X.
By [19, Example 7.1], 0 ∈ dom∂H. Hence (11) holds and thus z ∈ dom∂f. (cid:4)
3 Main Result
Theorem 3.1 Let A : X ⇉ X∗ be a maximal monotone linear relation, let f : X →
]−∞,+∞] be a proper lower semicontinuous sublinear function, and suppose that domA∩
intdom∂f 6= ∅. Then A+∂f is maximal monotone.
Proof. Let (z,z∗) ∈ X ×X∗ and suppose that
(18) (z,z∗) is monotonically related to gra(A+∂f).
By Fact 2.2, domA ⊆ P (F ) and dom∂f ⊆ P (F ). Hence,
X A X ∂f
(19) λ P (domF )−P (domF ) = X.
X A X ∂f
λ>0
[ (cid:0) (cid:1)
Thus, by Fact 2.3, it suffices to show that
(20) z ∈ domA∩dom∂f.
We have
hz,z∗i−hz,x∗i−hx,z∗i+hx,x∗i+hx−z,y∗i
(21) = hz −x,z∗ −x∗ −y∗i ≥ 0, ∀(x,x∗) ∈ graA,(x,y∗) ∈ gra∂f.
By Fact 2.6(ii), ∂f(0) 6= ∅. By (21),
inf[hz,z∗i−hz,A0i−hz,∂f(0)i] ≥ 0.
Thus,
(22) z ∈ X ∩(A0)⊥.
6
Then, by Fact 2.6(iii),
hz,z∗i ≥ f(z).
Thus,
(23) z ∈ domf.
By (22) and Lemma 2.9, we have
(24) z ∈ domA.
By Fact 2.6(i), y∗ ∈ ∂f(0) as y∗ ∈ ∂f(x). Then hx−z,y∗i ≤ f(x−z), ∀y∗ ∈ ∂f(x). Thus,
by (21), we have
(25) hz,z∗i−hz,x∗i−hx,z∗i+hx,x∗i+f(x−z) ≥ 0, ∀(x,x∗) ∈ graA,x ∈ dom∂f.
Let C := intdom f. Then by Fact 2.7, we have
(26) hz,z∗i−hz,x∗i−hx,z∗i+hx,x∗i+f(x−z) ≥ 0, ∀(x,x∗) ∈ graA,x ∈ C.
Set j := (f(·−z)+ιC)⊕ιX∗ and
(27) g: X ×X∗ → ]−∞,+∞] : (x,x∗) 7→ hx,x∗i+ι (x,x∗).
graA
By Fact 2.4, g is convex. Hence,
(28) h := g +j
is convex as well. Let
(29) c ∈ domA∩C.
0
By Lemma 2.8 and (23), z+c ∈ intdom f. Then there exists δ > 0 such that z+c +δB ⊆
0 0 X
domf and c + δB ⊆ domf. By (24), z + c ∈ domA since domA is a linear subspace.
0 X 0
Thus there exists b ∈ 1δB such that z+c +b ∈ domA∩intdomf. Let v∗ ∈ A(z+c +b).
2 X 0 0
Since c +b ∈ intdomf,
0
(30) (z +c +b,v∗) ∈ graA∩ intC ∩intdomf(·−z)×X∗ = domg ∩intdomj 6= ∅.
0
(cid:0) (cid:1)
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By Fact 2.1 and Fact 2.7, there exists (y∗,y∗∗) ∈ X∗ ×X∗∗ such that
h∗(z∗,z) = g∗(y∗,y∗∗)+j∗(z∗ −y∗,z −y∗∗)
= g∗(y∗,y∗∗)+ι (z −y∗∗)+sup[hx,z∗ −y∗i−f(x−z)]
{0}
x∈C
≥ g∗(y∗,y∗∗)+ι (z −y∗∗)+ sup [hx,z∗ −y∗i−f(x−z)] (by Lemma 2.8 and (23))
{0}
x∈z+C
= g∗(y∗,y∗∗)+ι (z −y∗∗)+hz,z∗ −y∗i+sup[hy,z∗ −y∗i−f(y)]
{0}
y∈C
= g∗(y∗,y∗∗)+ι (z −y∗∗)+hz,z∗ −y∗i+ sup [hky,z∗ −y∗i−f(ky)]
{0}
{y∈C,k>0}
= g∗(y∗,y∗∗)+ι (z −y∗∗)+hz,z∗ −y∗i+ sup k[hy,z∗ −y∗i−f(y)]
{0}
{y∈C,k>0}
(31)
≥ g∗(y∗,y∗∗)+ι (z −y∗∗)+hz,z∗ −y∗i.
{0}
By (26), we have, for every (x,x∗) ∈ graA∩(C×X∗), h(x,x∗),(z∗,z)i−h(x,x∗) = hx,z∗i+
hz,x∗i−hx,x∗i−f(x−z) ≤ hz,z∗i. Consequently,
(32) h∗(z∗,z) ≤ hz,z∗i.
Combining (31) with (32), we obtain
(33) g∗(y∗,y∗∗)+hz,z∗ −y∗i+ι (z −y∗∗) ≤ hz,z∗i.
{0}
Therefore, y∗∗ = z. Hence g∗(y∗,z)+hz,z∗ −y∗i ≤ hz,z∗i. Since g∗(y∗,z) = F (z,y∗), we
A
deduce that F (z,y∗) ≤ hz,y∗i. By Fact 2.2,
A
(34) (z,y∗) ∈ graA
Hence
z ∈ domA.
Apply Lemma 2.10 to obtain z ∈ dom∂f. Then z ∈ domA ∩ dom∂f. Hence A + B is
maximal monotone. (cid:4)
Remark 3.2 Verona and Verona (see [22, Corollary 2.9(a)] or [19, Theorem 53.1]) showed
the following: “Let f : X → ]−∞,+∞] be proper, lower semicontinuous, and convex, let
A : X ⇉ X∗ be maximal monotone, and suppose that dom A = X. Then ∂f+A is maximal
monotone.” Note that Theorem 3.1 cannot be deduced from this result because domA need
not have full domain.
8
Acknowledgment
Heinz Bauschke was partially supported by the Natural Sciences and Engineering Research
Council of Canada and by the Canada Research Chair Program. Xianfu Wang was partially
supported by the Natural Sciences and Engineering Research Council of Canada.
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