arXiv:1001.0257v1 [math.FA] 1 Jan 2010

On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function Heinz H. Bauschke∗, Xianfu Wang†, and Liangjin Yao‡ January 1, 2010

Abstract The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar’s constraint qualification holds. In this note, we provide a new maximal monotonicity result for the sum of a maximal monotone relation and the subdifferential operator of a proper, lower semicontinuous, sublinear function. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results on the Fitzpatrick function.

2000 Mathematics Subject Classification: Primary 47A06, 47H05; Secondary 47B65, 49N15, 52A41, 90C25 Keywords: Constraint qualification, convex function, convex set, Fenchel conjugate, Fitzpatrick function, linear relation, maximal monotone operator, multifunction, monotone operator, 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 gra A = (x, x∗ ) ∈ X × X ∗ | x∗ ∈ Ax be the graph of A. Recall that A is monotone if   (1) ∀(x, x∗ ) ∈ gra A ∀(y, y ∗ ) ∈ gra A hx − y, x∗ − y ∗ i ≥ 0, and maximal monotone if A is monotone and A has no proper monotone extension (in the sense of graph inclusion). We say A is a linear relation if gra A 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 (We also  the references therein. adopt standard notation used in these books: dom A = x ∈ X | Ax 6= ∅ is the domain of A. Given a subset C of X, int C is the interior of C, and C is the closure of C. We set 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 ιC , is defined at x ∈ X by ( 0, if x ∈ C; (2) ιC (x) := ∞, otherwise.

Given f : X → ]−∞, +∞], we set dom f = f −1 (R) and f ∗ : X ∗ → [−∞, +∞] : x∗ 7→ supx∈X (hx, x∗ i − f (x))  is the Fenchel conjugate of f . If f is convex and dom f 6= ∅, then ∂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) = λf (x) for  all x, y ∈ dom f and λ > 0. Finally, the closed unit ball in X is denoted by BX := x ∈ X | kxk ≤ 1 .) Throughout, we shall identify X with its canonical image in the bidual space X ∗∗ . Furthermore, X × X ∗ and (X × X ∗ )∗ = X ∗ × X ∗∗ are likewise paired via h(x, x∗ ), (y ∗ , y ∗∗ )i = hx, y ∗ i + hx∗ , y ∗∗ i, where (x, x∗ ) ∈ X × X ∗ and (y ∗ , y ∗∗ ) ∈ X ∗ × X ∗∗ . Let A and B be maximal monotone operators from X to X ∗ . Clearly, the sum 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 qualification dom A ∩ int dom B 6= ∅ when X is reflexive. The most famous open problem concerns the behaviour in nonreflexive Banach spaces. See Simons’ monograph  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  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 x0 ∈ dom f ∩ dom g such that g is continuous at x0 . Then for every z ∗ ∈ X ∗ , there exists y ∗ ∈ X ∗ such that (f + g)∗ (z ∗ ) = f ∗ (y ∗ ) + g ∗ (z ∗ − y ∗ ).

(3)

Furthermore, ∂(f + g) = ∂f + ∂g. Fact 2.2 (Fitzpatrick) (See [12, Corollary 3.9].) Let A : X ⇉ X ∗ be maximal monotone, and set  hx, a∗ i + ha, x∗ i − ha, a∗ i , (4) FA : X × X ∗ → ]−∞, +∞] : (x, x∗ ) 7→ sup (a,a∗ )∈gra A

which is the Fitzpatrick function associated with A. Then for every (x, x∗ ) ∈ X × X ∗ , the inequality hx, x∗ i ≤ FA (x, x∗ ) is true, and equality holds if and only if (x, x∗ ) ∈ gra A.

Fact 2.3 (Simons) S(See [19, Theorem 24.1(c)].) Let A, B : X ⇉ X ∗ be maximal monotone operators. Assume λ>0 λ [PX (dom FA ) − PX (dom FB )] is a closed subspace, where PX : (x, x∗ ) ∈ X × X ∗ → x. If (5)

(x, x∗ ) is monotonically related to gra(A + B) ⇒ x ∈ dom A ∩ dom B,

then A + B is maximal monotone. Fact 2.4 (Simons) (See [19, Lemma 19.7 and Section 22].) Let A : X ⇉ X ∗ be a monotone linear relation such that gra A 6= ∅. Then the function (6)

g : X × X ∗ → ]−∞, +∞] : (x, x∗ ) 7→ hx, x∗ i + ιgra A (x, x∗ )

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 inf f < λ < f (x) ≤ +∞, and set K :=

λ − f (a) . a∈X,a6=x kx − ak sup

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 ∈ dom f .

(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 int dom f 6= ∅. Then f is continuous on int dom f and ∂f (x) 6= ∅ for every x ∈ int dom f . Lemma 2.8 Let f : X → ]−∞, +∞] be a sublinear function. Then dom f + int dom f = int dom f . Proof. The result is trivial when int dom f = ∅ so we assume that x0 ∈ int dom f . Then there exists δ > 0 such that x0 + δBX ⊆ dom f . By sublinearity, ∀y ∈ dom f , we have y + x0 + δBX ⊆ dom f . Hence y + x0 ∈ int dom f. Then dom f + int dom f ⊆ int dom f . Since 0 ∈ dom f , int dom f ⊆ dom f + int dom f . Hence dom f + int dom f = int dom f .  Lemma 2.9 Let A : X ⇉ X ∗ be a maximal monotone linear relation, and let z ∈ X ∩(A0)⊥ . Then z ∈ dom A. Proof. Suppose to the contrary that z ∈ / dom A. Then the Separation Theorem provides ∗ ∗ w ∈ X such that (8)

hz, w∗ i > 0 and w∗ ∈ dom A . 4

Thus, (0, w∗ ) is monotonically related to gra A. Since A is maximal monotone, we deduce that w∗ ∈ A0. By assumption, hz, w∗ i = 0, which contradicts (8). Hence, z ∈ dom A.  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 dom A ∩ int dom ∂f 6= ∅, (z, z ∗ ) ∈ X × X ∗ is monotonically related to gra(A + ∂f ), and that z ∈ dom A. Then z ∈ dom ∂f . Proof. Let c0 ∈ X and y ∗ ∈ X ∗ be such that (9)

c0 ∈ dom A ∩ int dom ∂f

and (z, y ∗ ) ∈ gra A.

Take c∗0 ∈ Ac0 , and set  M := max ky ∗ k, kc∗0 k ,

(10)

D := [c0 , z], and h := f + ιD . By (9), Fact 2.7 and Fact 2.1, ∂h = ∂f + ∂ιD . Set H : X → ]−∞, +∞] : x 7→ h(x + z) − hz ∗ , xi. It remains to show that (11)

0 ∈ dom ∂H.

If inf H = H(0), then (11) holds. Now suppose that inf H < H(0). Let λ ∈ R be such that inf H < λ < H(0), and set λ − H(x) . kxk H(x)<λ

(12)

Kλ := sup

We claim that Kλ ≤ M. By Fact 2.5, we have Kλ ∈ ]0, ∞[ and ∀ε ∈ ]0, 1[, by gra ∂H = gra ∂h − (z, z ∗ ) there exists (x, x∗ ) ∈ gra ∂h such that (13)

hx − z, x∗ − z ∗ i ≤ −(1 − ε)Kλ kx − zk < 0.

Since ∂h = ∂f + ∂ιD , there exists t ∈ [0, 1] with x∗1 ∈ ∂f (x) and x∗2 ∈ ∂ιD (x) such that x = tc0 + (1 − t)z and x∗ = x∗1 + x∗2 . Then hx − z, x∗2 i ≥ 0. Thus, by (13), (14)

hx − z, x∗1 − z ∗ i ≤ hx − z, x∗1 + x∗2 − z ∗ i ≤ −(1 − ε)Kλ kx − zk < 0.

As x = tc0 + (1 − t)z and A is a linear relation, we have (x, tc∗0 + (1 − t)y ∗ ) ∈ gra A. Since (z, z ∗ ) is monotonically related to gra(A + ∂f ), by (10), (15)

hx − z, x∗1 − z ∗ i ≥ −hx − z, tc∗0 + (1 − t)y ∗ i ≥ −M kx − zk. 5

Combining (15) and (14), we obtain (16)

−M kx − zk ≤ −(1 − ε)Kλ kx − zk < 0.

Hence, (1 − ε)Kλ ≤ M . Letting ε ↓ 0, we deduce that Kλ ≤ M . Then, by (12) and letting λ ↑ H(0), we get (17)

H(y) + M kyk ≥ H(0),

∀y ∈ X.

By [19, Example 7.1], 0 ∈ dom ∂H. Hence (11) holds and thus z ∈ dom ∂f .

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 dom A ∩ int dom ∂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, dom A ⊆ PX (FA ) and dom ∂f ⊆ PX (F∂f ). Hence, [  (19) λ PX (dom FA ) − PX (dom F∂f ) = X. λ>0

Thus, by Fact 2.3, it suffices to show that (20)

z ∈ dom A ∩ dom ∂f.

We have

(21)

hz, z ∗ i − hz, x∗ i − hx, z ∗ i + hx, x∗ i + hx − z, y ∗ i = hz − x, z ∗ − x∗ − y ∗ i ≥ 0, ∀(x, x∗ ) ∈ gra A, (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 ∈ dom f.

By (22) and Lemma 2.9, we have z ∈ dom A.

(24)

By Fact 2.6(i), y ∗ ∈ ∂f (0) as y ∗ ∈ ∂f (x). Then hx − z, y ∗ i ≤ f (x − z), by (21), we have (25)

hz, z ∗ i − hz, x∗ i − hx, z ∗ i + hx, x∗ i + f (x − z) ≥ 0,

∀y ∗ ∈ ∂f (x). Thus,

∀(x, x∗ ) ∈ gra A, x ∈ dom ∂f.

Let C := int dom 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∗ ) ∈ gra A, x ∈ C.

Set j := (f (· − z) + ιC ) ⊕ ιX ∗ and (27)

g : X × X ∗ → ]−∞, +∞] : (x, x∗ ) 7→ hx, x∗ i + ιgra A (x, x∗ ).

By Fact 2.4, g is convex. Hence, (28)

h := g + j

is convex as well. Let (29)

c0 ∈ dom A ∩ C.

By Lemma 2.8 and (23), z +c0 ∈ int dom f . Then there exists δ > 0 such that z +c0 +δBX ⊆ dom f and c0 + δBX ⊆ dom f . By (24), z + c0 ∈ dom A since dom A is a linear subspace. Thus there exists b ∈ 21 δBX such that z + c0 + b ∈ dom A ∩ int dom f . Let v ∗ ∈ A(z + c0 + b). Since c0 + b ∈ int dom f ,  (30) (z + c0 + b, v ∗ ) ∈ gra A ∩ int C ∩ int dom f (· − z) × X ∗ = dom g ∩ int dom j 6= ∅.

7

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{0} (z − y ∗∗ ) + sup [hx, z ∗ − y ∗ i − f (x − z)] x∈C ∗

∗∗

∗∗

≥ g (y , y ) + ι{0} (z − y ) + sup [hx, z ∗ − y ∗ i − f (x − z)] (by Lemma 2.8 and (23)) ∗

∗∗

∗∗

x∈z+C ∗

∗∗

∗∗

∗∗

∗∗

= g (y , y ) + ι{0} (z − y ) + hz, z − y ∗ i + sup [hy, z ∗ − y ∗ i − f (y)] y∈C

= g (y , y ) + ι{0} (z − y ) + hz, z − y i +

sup

[hky, z ∗ − y ∗ i − f (ky)]

{y∈C,k>0}

= g (y , y ) + ι{0} (z − y ) + hz, z − y i +

sup

k [hy, z ∗ − y ∗ i − f (y)]

{y∈C,k>0}

(31) ≥ g ∗ (y ∗ , y ∗∗ ) + ι{0} (z − y ∗∗ ) + hz, z ∗ − y ∗ i. By (26), we have, for every (x, x∗ ) ∈ gra A ∩ (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{0} (z − y ∗∗ ) ≤ hz, z ∗ i.

Therefore, y ∗∗ = z. Hence g ∗ (y ∗ , z) + hz, z ∗ − y ∗ i ≤ hz, z ∗ i. Since g ∗ (y ∗ , z) = FA (z, y ∗ ), we deduce that FA (z, y ∗ ) ≤ hz, y ∗ i. By Fact 2.2, (34)

(z, y ∗ ) ∈ gra A

Hence z ∈ dom A. Apply Lemma 2.10 to obtain z ∈ dom ∂f . Then z ∈ dom A ∩ dom ∂f . Hence A + B is maximal monotone.  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 dom A 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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