arXiv:0911.2001v1 [math.FA] 10 Nov 2009

An affirmative answer to a problem posed by Z˘alinescu Liangjin Yao∗. November 9, 2009

Abstract Recently, in [5] Zˇ alinescu posed a question about the characterization of the intrinsic core of the Minkowski sum of two graphs associated with two maximal monotone operators. In this note we give an affirmative answer.

2000 Mathematics Subject Classification: Primary 47H05; Secondary 49J53, 52A41. Keywords: Convex function, convex set, Fenchel conjugate, Fitzpatrick function, intrinsic core, maximal monotone operator, monotone operator, multifunction, relative algebraic interior, representative, set-valued operator.

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Introduction

We suppose throughout this note that X is a real reflexive Banach space with norm k · k and dual product h·, ·i. We now introduce some notation. Let A : X ⇉ X ∗ be a set-valued operator or multifunction whose graph is defined by  gra A := (x, x∗ ) ∈ X × X ∗ | x∗ ∈ Ax .  The domain of A is dom A := x ∈ X | Ax 6= ∅ . Recall that A is monotone if for all (x, x∗ ), (y, y ∗ ) ∈ gra A we have hx − y, x∗ − y ∗ i ≥ 0, and A is maximal monotone if A is monotone and A has no proper monotone extension (in the sense of graph inclusions). ∗ Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail: [email protected].

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The Fitzpatrick function of A (see [1]) is given by FA : (x, x∗ ) 7→

(1)

sup (a,a∗ )∈gra A


hx, a∗ i + ha, x∗ i − ha, a∗ i .

For a function f : X → ]−∞, +∞], the domain is dom f := {x ∈ X | f (x) < +∞} and f ∗ : X ∗ → [−∞, +∞] : x∗ 7→ supx∈X (hx, x∗ i − f (x)) is the Fenchel conjugate of f . Given F : X × X ∗ → ]−∞, +∞], we say F is a representative of a maximal monotone operator A if F is lower semicontinuous and convex with F ≥ h·, ·i, F ∗ ≥ h·, ·i and gra A = {(x, x∗ ) | F (x, x∗ ) = hx, x∗ i}. Following [2], it will be convenient to set F ⊺ : X ∗ × X :→ ]−∞, +∞] : (x∗ , x) 7→ F (x, x∗ ), where F : X × X ∗ → ]−∞, +∞], and similarly for a function defined on X ∗ × X. We define Fb (see [5]) by

Fb (x, x∗ ) := F (x, −x∗ ).

Let a = (x, x∗ ), b = (y, y ∗ ) ∈ X × X ∗ , we also set (see [4]) by ⌊a, b⌋ = hx, y ∗ i + hy, x∗ i. Given a subset D of X, D is the closure, conv D is the convex hull, and aff D is the affine hull. The conic hull of D is denoted by cone D := {λx | λ ≥ 0, x ∈ D}. The indicator function ιD : X → ]−∞, +∞] of D is defined by ( 0, if x ∈ D; x 7→ +∞, otherwise. The intrinsic core or relative algebraic interior of D, written as i D in [6], is i

D := {a ∈ D | ∀x ∈ aff(D − D), ∃δ > 0, ∀λ ∈ [0, δ] : a + λx ∈ D}.

We define ic D by ic

D :=

(

i D,

∅,

if aff D is closed; otherwise.

Zˇ alinescu posed the following problem in [5]: Let A, B : X ⇉ X ∗ be maximal monotone. Is the implication i h ic [conv (gra A − gra(−B))] 6= ∅ ⇒ ic dom FA − dom Fc B 6= ∅

true? Theorem 2.7 provides an affirmative answer to this question. It further shows that these two sets actually are equal. 2

2

Main result

Definition 2.1 (Fitzpatrick family) Let A : X ⇉ X ∗ be a maximal monotone operator. The associated Fitzpatrick family FA consists of all functions F : X × X ∗ → ]−∞, +∞] that are lower semicontinuous and convex, and that satisfy F ≥ h·, ·i, and F = h·, ·i on gra A. Fact 2.2 (Fitzpatrick) (See [1, Theorem 3.10].) Let A : X ⇉ X ∗ be a maximal monotone operator. Then for every (x, x∗ ) ∈ X × X ∗ ,   (2) FA (x, x∗ ) = min F (x, x∗ ) | F ∈ FA and FA∗⊺(x, x∗ ) = max F (x, x∗ ) | F ∈ FA .

Fact 2.3 (Simons) (See [4, Lemma 20.4(b)].) Let A : X ⇉ X ∗ be maximal monotone and a := (x, x∗ ) ∈ X × X ∗ with hx, x∗ i = 0. Suppose that there exists u ∈ R such that ⌊gra A, a⌋ = {u}.

Then ⌊dom FA , a⌋ = {u}. Theorem 2.4 Let A, B : X ⇉ X ∗ be maximal monotone. Then i h (3) aff [gra A − gra(−B)] = aff dom FA − dom Fc B .

Proof. We do and can suppose (0, 0) ∈ gra A and (0, 0) ∈ gra B. We first show i h dom FA − dom Fc (4) B ⊆ aff [gra A − gra(−B)].

i h Suppose to the contrary that there exists c ∈ X × X ∗ such that c ∈ dom FA − dom Fc B but c ∈ / aff [gra A − gra(−B)]. By the Separation Theorem, there exist a := (x, x∗ ) ∈ X × X ∗ and δ ∈ R such that  (5) ⌊a, c⌋ > δ > sup ⌊a, e⌋ | e ∈ aff [gra A − gra(−B)] .

Since (0, 0) ∈ gra A, (0, 0) ∈ gra B and aff [gra A − gra(−B)] is a closed subspace, we have δ > 0 and ⌊a, b − d⌋ = 0, ∀b ∈ gra A, ∀d ∈ gra(−B). Thus, (6)

⌊a, gra A⌋ = {0} = ⌊a, gra(−B)⌋ = ⌊(−x, x∗ ), gra B⌋.

By (0, 0) ∈ gra A and (0, 0) ∈ gra B again, (7)

FA (a) = FA (x, x∗ ) = 0,

FB (−x, x∗ ) = 0.

Since FA (x, x∗ ) ≥ hx, x∗ i and FB (−x, x∗ ) ≥ h−x, x∗ i, by (7), hx, x∗ i = 0. Thus by (6) and Fact 2.3, ⌊a, dom FA ⌋ = {0} = ⌊(−x, x∗ ), dom FB ⌋ = ⌊a, dom Fc B ⌋. 3

Thus, ⌊a, dom FA − dom Fc B ⌋ = {0}, which contradicts (5). Hence i h dom FA − dom Fc (8) B ⊆ aff [gra A − gra(−B)]. i h And thus aff dom FA − dom Fc B ⊆ aff [gra A − gra(−B)]. Hence

i h aff [gra A − gra(−B)] = aff dom FA − dom Fc B .



Fact 2.5 (Zˇ alinescu) (See [5, Lemma 2 and Theorem 3].) Let A, B : X ⇉ X ∗ be maximal monotone, and let F1 , F2 be representatives of A, B, respectively. Then ic

[gra A − gra(−B)] = ic [conv(gra A − gra(−B))]

and ic

If

ic

(9)

h

h

i i h c2 . c2 ⊆ [gra A − gra(−B)] ⊆ conv [gra A − gra(−B)] ⊆ dom F1 − dom F dom F1 − dom F

i c2 = 6 ∅, then dom F1 − dom F ic

h

i c2 = ic [gra A − gra(−B)] = ic [conv(gra A − gra(−B))]. dom F1 − dom F

Remark 2.6 If X is finite-dimensional, the intrinsic core of a convex h set D ⊆ X is ithe c2 = same as the relative interior of D in the sense of Rockafellar[3]. Then ic dom F1 − dom F i h i dom F − dom F c2 6= ∅ by [3, Theorem 6.2]. Thus, (9) always holds. 1

Our main result comes the following which provides an affirmative answer to the question posed by Z˘ alinescu.

Theorem 2.7 Let A, B : X ⇉ X ∗ be maximal monotone such that ic [conv (gra A − gra(−B))] 6= ∅. Then i h ic [gra A − gra(−B)] = ic [conv (gra A − gra(−B))] = ic dom FA − dom Fc B .

Moreover, if F1 , F2 are representatives of A, B, respectively, then i h i h ic c2 = ic [conv (gra A − gra(−B))] = ic dom FA − dom Fc (10) dom F1 − dom F B .

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i h and Proof. Let a ∈ ic [conv (gra A − gra(−B))]. Then we have a ∈ dom FA − dom Fc B cone [conv (gra A − gra(−B)) − a] is a closed subspace. By Theorem 2.4, h i cone [conv (gra A − gra(−B)) − a] ⊆ cone dom FA − dom Fc B −a i h i h c ⊆ aff dom FA − dom Fc B − a = aff dom FA − dom FB − {a} ⊆ aff [gra A − gra(−B)] − {a} ⊆ aff [gra A − gra(−B) − a]

= aff [conv (gra A − gra(−B)) − a] ⊆ cone [conv (gra A − gra(−B)) − a] . i Hence cone dom FA − dom Fc − a = cone [conv (gra A − gra(−B)) − a] is a closed subspace. B i h Thus a ∈ ic dom FA − dom Fc B . By Fact 2.5, i h ic [gra A − gra(−B)] = ic [conv (gra A − gra(−B))] = ic dom FA − dom Fc B . h

And by Fact 2.2,

i i h h c2 ⊆ dom FA − dom Fc conv [gra A − gra(−B))] ⊆ dom F1 − dom F B .

Similar to the proof above, see that (10) holds.



Theorem 2.8 Let A, B : X ⇉ X ∗ be maximal monotone, and F1 , F2 be representatives of A, B, respectively. Then i h ic c2 = ic [conv (gra A − gra(−B))] = ic [gra A − gra(−B)]. (11) dom F1 − dom F Proof. We consider two cases.

i h c2 6= ∅. Then by gra(−B))] = ∅. Assume that ic dom F1 − dom F i h c2 = ic [conv (gra A − gra(−B))] = ∅. This a contradiction. Fact 2.5, ic dom F1 − dom F Case 1:

Case 2:

ic [conv (gra A −

ic [conv (gra A −

gra(−B))] 6= ∅. Apply Theorem 2.7.

Combining the above results, we see that (11) holds.  Corollary 2.9 Let A, B : X ⇉ X ∗ be maximal monotone, and F1 , F2 be representatives of A, B, respectively. Assume ic [conv (gra A − gra(−B))] 6= ∅. Then i h c2 = conv [(gra A − gra(−B)] = [gra A − gra(−B)]. dom F1 − dom F (12) In particular,

h

i dom FA − dom Fc B = conv [(gra A − gra(−B)] = [gra A − gra(−B)]. 5

Proof. Given a convex set D ⊆ X, assume that ic D 6= ∅, then ic D = D. By Theorem 2.8, h

i c2 = conv [(gra A − gra(−B)] = ic [gra A − gra(−B)] ⊆ [gra A − gra(−B)] dom F1 − dom F

Hence (12) holds.



Acknowledgment The author thanks Heinz Bauschke and Xianfu Wang for valuable discussions.

References [1] S. Fitzpatrick, “Representing monotone operators by convex functions”, in Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 20, Canberra, Australia, pp. 59–65, 1988. [2] J.-P. Penot, “The relevance of convex analysis for the study of monotonicity”, Nonlinear Analysis, vol. 58, pp. 855–871, 2004. [3] R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [4] S. Simons, From Hahn-Banach to Monotonicity, Springer-Verlag, 2008. [5] C. Zˇ alinescu, “A new convexity property for monotone operators”, Journal of Convex Analysis, vol. 13, pp. 883–887, 2006. [6] C. Z˘ alinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002.

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