Partially Ball Weakly Inf-Compact Saddle Functions

MATHEMATICS OF OPERATIONS RESEARCH Vol. 30, No. 2, May 2005, pp. 404–419 issn 0364-765X eissn 1526-5471 05 3002 0404

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doi 10.1287/moor.1040.0126 © 2005 INFORMS

Partially Ball Weakly Inf-Compact Saddle Functions Lionel Thibault

Université Montpellier II, Département de Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France, [email protected]

Nadia Zlateva

Section of Operations Research, Department of Mathematics, Sofia University, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria, [email protected]fia.bg, and Université Montpellier II, Département de Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France, [email protected] We study on a product Banach space the properties of a class of saddle functions called partially ball weakly inf-compact. For such a function we prove that the domain of the subdifferential is nonempty, that the operator naturally associated with the subdifferential is maximal monotone, and that the subdifferential of the function is integrable. For a function in a large subclass of that class we prove the density of the domain of the subdifferential in the domain of the function. Key words: saddle function; subdifferential; integrability; maximal monotone operator MSC2000 subject classification: Primary: 49J52; secondary: 28B20 OR/MS subject classification: Primary: Convexity/functions History: Received September 30, 2003; revised June 18, 2004.

1. Introduction. The aim of the present paper is to study saddle functions K X × Y → − + defined on a product of Banach spaces X and Y . Such functions are closely related to minimax problems. The main contribution in the study of saddle functions with values in − + is due to Rockafellar. In the case when X and Y are finite dimensional, their properties are investigated in detail in a number of works of Rockafellar (cf., e.g., [16, 18, 19]), McLinden [9, 10, 11], etc. Most of those properties are generalized to the case when X and Y are Banach spaces, one of which is reflexive, in subsequent papers of Rockafellar [20, 22], Gossez [6], etc. Our intention here is to extend their results to a class of saddle functions defined on a product of two arbitrary Banach spaces X and Y . We begin by giving some necessary preliminaries and notation in §2. The properties of saddle functions on the Banach space X × Y are laid out in §3. Section 4 is devoted to the study of subdifferentiability properties of saddle functions, and more precisely, the subdifferential properties of a proper closed saddle function K X × Y → − + that we call partially ball weakly inf-compact (Definition 4.1) and write pbwc for short. We establish that the domain of the subdifferential K is nonempty (Theorem 4.1). Moreover, in this setting the operator TK associated with K is maximal monotone (Theorem 4.3). For proper closed saddle functions K in a large subclass of pbwc saddle functions we show that the domain of K is dense in the domain of K (Theorem 4.2). In the final section, §5, we prove that for proper closed pbwc saddle functions K the subdifferential K is integrable (Theorems 5.1 and 5.2). We can conclude that there exist clear parallels between basic properties of proper closed convex functions defined on a Banach space X and those of proper closed pbwc saddle functions defined on a product Banach space X × Y . 2. Notation and preliminaries. Let us start with a brief review of the terminology. We will denote by X a real Banach space and X ∗ will stand for its topological dual space, i.e., the set of continuous linear functionals on X. If x∗ ∈ X ∗ , we will write x x∗ for the value of x∗ at x ∈ X. Recall that the weak topology w X X ∗ is the smallest topology on X with respect to which all the functions · x∗ (x∗ ∈ X ∗ ) are continuous and that the weak-star topology w X ∗ X is the smallest topology on X ∗ with respect to which all the 404

Thibault and Zlateva: Partially Ball Weakly Inf-Compact Saddle Functions Mathematics of Operations Research 30(2), pp. 404–419, © 2005 INFORMS

405

functions x · (x ∈ X) are continuous. The dual of the Banach space X ∗ is called the bidual of X and it is denoted by X ∗∗ . Any element of X “gives” an element of X ∗∗ via the canonical embedding X → X ∗∗ defined by x∗ x

ˆ = x x∗

for x ∈ X and x∗ ∈ X ∗

Let us recall that two normed linear spaces are called congruent (relation denoted by ) if there exists a norm-preserving isomorphism (called a congruence) between them. It is well known (see Holmes [7]) that the canonical embedding is a congruence between X and The image X of X is a norm closed subspace of X ∗∗ in X ∗∗ , i.e., X X. its image X ∗∗ coincides with X , the Banach space X is said to be reflexive. and x ˆ = x. When X ∗ , i.e., X ∗ X ∗ via the canonical Analogously, the Banach space X ∗ is congruent to X ∗ ∗∗∗ embedding X → X defined by x∗∗ x∗ = x∗ x∗∗

for x∗ ∈ X ∗ and x∗∗ ∈ X ∗∗

∗ X ∗ (see Holmes [7, p. 123]). Moreover, X A convex function f on X is an everywhere defined function with values in the extended real interval − + whose epigraph epi f = x r ∈ X × f x ≤ r is a convex set in X × . The effective domain of f is defined by dom f = x ∈ X f x < +. If f x > − for all x and f x < + for at least one x, then f is said to be proper. Otherwise, f is said to be improper. The convex function f is said to be closed if it is proper and lower semicontinuous, or else, if it is identically + or −. Through the paper, unless otherwise specified, the closure and lower semicontinuity operations will be taken with respect to the norm topology. Given any convex function f on X, there exists a greatest closed convex function majorized by f . This function is called the closure of f and is denoted by cl f . It is clear that cl f ≤ f and f is closed exactly when f = cl f . When f does not take the value −, then cl f x = lim inf f x x →x

for all x ∈ X

(2.1)

For any convex function f on X, the function f ∗ X ∗ → − + defined by f ∗ x∗ = sup x x∗ − f x x∈X

is called the conjugate of f . The conjugate of a proper lower semicontinuous convex function f on X is a proper convex function on X ∗ which is lower semicontinuous with respect to the weak-star topology w X ∗ X , as well as to the norm topology of X ∗ . One defines the biconjugate of a convex function f on X as the conjugate of its conjugate function, i.e., it is the function f ∗∗ X ∗∗ → − + on the bidual space X ∗∗ defined by f ∗∗ x∗∗ = sup x∗ x∗∗ − f ∗ x∗ x∗ ∈X ∗

The biconjugate of a proper convex lower semicontinuous function f is a proper convex function on X ∗∗ which is lower semicontinuous with respect to the w X ∗∗ X ∗ topology, as well as to the norm topology of X ∗∗ . By Fenchel’s duality result, for a convex function f one has ˆ = cl f x for all x ∈ X (2.2) f ∗∗ x The reader interested in the theory of conjugate convex functions could consult for example Brøndsted [2], Fenchel [5], Moreau [12, 13], and Rockafellar [17, 19]. A subgradient of a convex function f at a point x is an element x∗ ∈ X ∗ such that f x ≥ f x + x − x x∗

∀ x ∈ X

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The (possibly empty) set of all such subgradients of f at x is denoted by f x and the ∗ multivalued mapping f X → 2X is called the subdifferential of f . The domain of f is defined by dom f = x ∈ X f x = . Obviously, when f is proper, dom f ⊂ dom f . We will often use the following well-known result (see Brøndsted and Rockafellar [3]): If the convex function f X → ∪ + is proper and lower semicontinuous, then dom f is dense in dom f

(2.3)

Recall that the range of f is the subset of X ∗ given by Rge f = x∈X f x while the graph of f is the set gph f = x x∗ ∈ X × X ∗ x∗ ∈ f x . It is well known (see for example Aubin and Ekeland [1], Moreau [13], and Rockafellar [19]) that for a proper convex function f , the following are equivalent: x∗ ∈ f x ⇐⇒ f x + f ∗ x∗ = x x∗ and any of those implies that x ∈ dom f . If, in addition, f is lower semicontinuous at x, then x∗ ∈ f x ⇐⇒ f x + f ∗ x∗ = x x∗ ⇐⇒ xˆ ∈ f ∗ x∗

(2.4)

Following Rockafellar [19, Corollary 23.5.2], one derives that if a proper convex function f is subdifferentiable at x then cl f x = f x and cl f x = f x

(2.5)

If X is reflexive and f is a proper lower semicontinuous convex function, from (2.4) it is clear that one can identify f ∗∗ with f and that f ∗ is just the “inverse” of f . In other words, x ∈ f ∗ x∗ if and only if x∗ ∈ f x . If X is not reflexive, the relationship between f ∗ and f is more complicated, but f ∗ and f still completely determine each other, according to the following result due to Rockafellar [21]. Theorem 2.1 (Rockafellar [21, Proposition 1]). Let f X → ∪ + be a proper lower semicontinuous convex function and let x∗ ∈ X ∗ , x∗∗ ∈ X ∗∗ . Then x∗∗ ∈ f ∗ x∗ if and only if there exists a net x∗ ∈A in X ∗ converging to x∗ in the norm topology and a (with the same partially ordered index set A) converging to x∗∗ bounded net x ∈A in X ∗∗ ∗ in the w X X topology such that x∗ ∈ f x for every ∈ A. For any r ∈ , the r-sublevel set of the convex function f is the (possibly empty) set f ≤ r = x ∈ X f x ≤ r. Obviously, it is a convex set and when f is lower semicontinuous, it is closed in the norm topology as well as in the weak topology of X. Let BX denote the closed unit ball of the Banach space X. We say that the function f X → ∪ + is ball weakly inf-compact (bwc for short) if for any r ∈ the sets Levr n f = f ≤ r ∩ nBX are w X X ∗ compact for any n ∈ . We make the convention that the empty set is weakly compact. Recall that the notion of inf-compactness was introduced by Moreau (see [13]). From Theorem 2.1 we derive the following characterisations of a proper closed bwc convex function. Theorem 2.2. Let f X → ∪ + be a proper lower semicontinuous convex function. Then the following are equivalent: (a) f is bwc; (b) the range of f ∗ is a nonempty subset of X; (c) dom f ∗∗ ⊂ X.


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Proof. (a) ⇒ (b). Since f ∗ is a proper closed convex function, the range of f ∗ is nonempty according to (2.3). Take any x∗∗ ∈ f ∗ x∗ . Then by Theorem 2.1 we have that there exist a net x∗ ∈A in X ∗ converging to x∗ in the norm topology and a bounded net converging to x∗∗ in the w X ∗∗ X ∗ topology such that x∗ ∈ f x for every x ∈A in X ∈ A. As f is proper we take any x0 ∈ dom f and by the definition of the subdifferential we have f x ≤ f x0 + x − x0 x∗ ∀ ∈ A If we set r = 1 + f x0 + x∗ x∗∗ − x0 , there exists some 0 ∈ A such that f x ≤ r for all ≥ 0 . Since the net x ∈A is norm bounded, for sufficiently large n ∈ , the points x ∈ f ≤ r ∩ nBX for all ≥ 0 . The w X X ∗ compactness of the latter set ensures that (cf., e.g., Holmes [7, p. 149]). the w X ∗∗ X ∗ closure of its embedding in X ∗∗ lies in X ˆ Hence, there exists some x ∈ X such that x∗∗ = x. (b) ⇒ (c). Suppose the contrary, i.e., there exists some x∗∗ ∈ dom f ∗∗ such that x∗∗ ∈ ∗∗ = . X \X. Then there exists a norm open neighbourhood U of x∗∗ in X ∗∗ such that U ∩ X By the norm density of the domain of the subdifferential of a proper closed convex function in its effective domain (see (2.3)) it follows the existence of some x0∗∗ ∈ U ∩ dom f ∗∗ . Let x0∗∗∗ ∈ f ∗∗ x0∗∗ . By Theorem 2.1 we have that there exist a net x∗∗ ∈A in X ∗∗ converging ∗ converging to x0∗∗∗ to x0∗∗ in the norm topology of X ∗∗ and a bounded net x∗ ∈A in X ∗∗∗ ∗∗ ∗∗ ∗ ∗ in the w X X topology such that x ∈ f x for every ∈ A. From the norm convergence of x∗∗ to x0∗∗ we have that x∗∗ are eventually in U , in particular x∗∗ ∈ X. ∗∗ ∗∗ However, from (b) we have that x ∈X, which yields a contradiction. Hence, dom f ⊂ X. (c) ⇒ (a). Let us fix r ∈ and n ∈ such that Levr n f is a nonempty set; otherwise the claim is trivial. Take an arbitrary net x ∈A ⊂ Levr n f . Since x = x and f x = f ∗∗ x by (2.2), we have x ∈ S = x∗∗ ∈ X ∗∗ f ∗∗ x∗∗ ≤ r ∩ nBX ∗∗ . The later set being obviously w X ∗∗ X ∗ compact one may extract from x ∈A a subnet xˆs ∈ that converges to some x∗∗ ∈ S in the w X ∗∗ X ∗ topology. The definition of S yields that x∗∗ ∈ dom f ∗∗ By the assumption of (c) we have that x∗∗ = xˆ for some x ∈ X, and x∗∗ ∈ nBX ∗∗ implies that x ∈ nBX Obviously, xs tends to x in the w X X ∗ topology and x ∈ Levr n f since f x = f ∗∗ x ˆ by (2.2). From the arbitrary net x ∈A ⊂ Levr n f we have shown how to obtain a subnet that is w X X ∗ convergent to an element of Levr n f . This means that the latter set is w X X ∗ compact. The proof is then complete. To round off this section, let us recall that a concave function g on X is an everywhere defined function with values in the extended real interval − + such that the function −g is convex. 3. Saddle functions. Properties. Let us consider a space Z, which is a product of two real Banach spaces X and Y , i.e., Z = X × Y . Setting any reasonable norm on X × Y we have that Z is a Banach space (take for instance x y = maxx y) and its dual can be identified with X ∗ × Y ∗ using the pairing x y x∗ y ∗ = x x∗ + y y ∗ . A saddle function on Z is an everywhere defined function K with values in − + such that K · y is convex on X for each y ∈ Y and K x · is concave on Y for each x ∈ X. We denote by cl1 K the saddle function obtained by closing K x y as a convex function of x for each fixed y, i.e., cl1 K x y = cl K · y x . Similarly, we denote by cl2 K the saddle function obtained by closing −K x y as a convex function of y for each fixed x and after that by taking its negative, i.e., cl2 K x y = −cl −K x · y . Clearly, cl1 K ≤ K ≤ cl2 K. Two saddle functions K and L on Z are said to be equivalent (written K ∼ L) if cl1 K = cl1 L and cl2 K = cl2 L (see Gossez [6] and Rockafellar [19, 22]). When K ∼ L, we say that K and L belong to the same equivalence class and that K and L are representatives of the class. There exists another definition for the relation K ∼ L which is, of course, equivalent to the former. For the second one we need to introduce two more notions.

Thibault and Zlateva: Partially Ball Weakly Inf-Compact Saddle Functions

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Mathematics of Operations Research 30(2), pp. 404–419, © 2005 INFORMS

The function on X × Y ∗ obtained by taking the conjugate of −K x y in the second argument (its convex argument) when the first argument is fixed, i.e., F x · = −K x · ∗ , or F x y ∗ = supK x y + y y ∗ (3.1) y∈Y

will be called the convex parent of K (see Rockafellar [18]). It is a convex function over X × Y ∗ . Dually, the concave parent of K is defined by G · y = −K · y ∗ , or G x∗ y = inf K x y − x x∗ x∈X

(3.2)

All parent functions that appear in the paper are considered as functions of the joint variable belonging to the product Banach space, and their effective domains and subdifferentials are taken in this setting. It is shown by Rockafellar (see [19]) that two saddle functions are equivalent exactly when they have the same parent functions. Here we give a proof for completeness and for the convenience of the reader. Lemma 3.1. Let K L X × Y → − + be two saddle functions. Then K ∼ L if and only if they have the same parent functions. Proof. First, let us suppose that K ∼ L, i.e., cl1 K = cl1 L and cl2 K = cl2 L according to the original definition. Denote by F the convex parent of K and by F the convex parent of L. Then F x y ∗ = supK x y + y y ∗ = supcl2 K x y + y y ∗ y∈Y

y∈Y

(3.3)

where for the latter equality we use the fact that a convex function and its closure have the same conjugate function (see Moreau [13]). Hence, F x y ∗ = supcl2 K x y + y y ∗ = supcl2 L x y + y y ∗ = F x y ∗ y∈Y

y∈Y

Analogously, for the concave parent G of K and the concave parent G of L one obtains that G x∗ y = G x∗ y . Now, let us suppose that F = F and G = G . From Fenchel duality (2.2) we have −cl2 K x y = sup y y ∗ − F x y ∗ = sup y y ∗ − F x y ∗ = −cl2 L x y % y ∗ ∈Y ∗

y ∗ ∈Y ∗

hence cl2 K = cl2 L. Analogously, cl1 K = cl1 L and the proof is complete. If cl1 K and cl2 K are both equivalent to K (written K ∼ cl1 K ∼ cl2 K), then K is said to be a closed saddle function (see Gossez [6] and Rockafellar [19, 22]). A necessary and sufficient condition for K to be closed is cl1 cl2 K = cl1 K

and

cl2 cl1 K = cl2 K

(3.4)

It is easy to see that when K is a closed saddle function and L is a saddle function equivalent to K, i.e., L ∼ K, then L is closed too. The effective domain of a saddle function K (see Rockafellar [18, 19, 20]) is defined as the set Dom K = CK × DK , where CK = x ∈ X K x y < + ∀ y ∈ Y DK = y ∈ Y K x y > − ∀ x ∈ X A basic disadvantage of this definition comes from the fact that Dom K depends on the representative of the equivalence class of K, as it can be seen from an example due to


409


Rockafellar (see Gossez [6]). Hence, one introduces the following more appropriate notion for the domain of a saddle function K that depends only on the equivalence class to which K belongs (and not on the representatives of the class). The domain of a saddle function K (see Gossez [6] and Rockafellar [22]) is defined by Dom K = CK × DK , where CK = x ∈ X cl2 K x y < + ∀ y ∈ Y DK = y ∈ Y cl1 K x y > − ∀ x ∈ X Clearly, Dom K ⊂ Dom K. If K is closed, then Dom K is dense in Dom K, and Dom K = Dom cl1 K ∩ Dom cl2 K (see Gossez [6, Proposition 1]). The saddle function K is said to be proper if Dom K is a nonempty set. From the preceding, if K is closed, it is proper exactly when Dom K is a nonempty set. Lemma 3.2.

Let K X × Y → − + be a saddle function. Then

Dom K = x y ∈ X × Y − < cl1 K x y ≤ K x y ≤ cl2 K x y < + Proof. Let us denote V = x y ∈ X × Y − < cl1 K x y ≤ K x y ≤ cl2 K x y < + The inclusion V ⊂ Dom K being obvious for V = , we may suppose that V = . Take any x0 y0 ∈ V . Suppose that y0 ∈ DK . Then there exists x˜ ∈ X such that cl1 K x ˜ y0 = −. Then, by the definition of the closure of a convex function, it follows that cl1 K x y0 = − for all x ∈ X, which yields a contradiction, since cl1 K x0 y0 is finite. Hence, y0 ∈ DK . Analogously, one shows that x0 ∈ CK and then x0 y0 ∈ Dom K, so the inclusion V ⊂ Dom K is established. The opposite inclusion, i.e., Dom K ⊂ V , being obvious for Dom K = , we suppose it is nonempty and take any x0 y0 ∈ Dom K. Then x0 ∈ CK

=⇒

cl2 K x0 y < + ∀ y ∈ Y

=⇒

cl2 K x0 y0 < +

y0 ∈ DK

=⇒

cl1 K x y0 > − ∀ x ∈ X

=⇒

cl1 K x0 y0 > −%

hence x0 y0 ∈ V . The proof is then complete. The following statement concerns a useful property of a proper closed saddle function. Lemma 3.3. Let K X × Y → − + be a proper closed saddle function. Then (a) for any y ∈ DK , the function cl1 K · y is a proper lower semicontinuous convex function with dom cl1 K · y ⊃ CK and, for any y ∈ DK , the function cl1 K · y = −; (b) for any x ∈ CK , the function −cl2 K x · is a proper lower semicontinuous convex function with dom −cl2 K x · ⊃ DK and, for any x ∈ CK , the function −cl2 K x · = −. Proof. We will prove (a), the proof of (b) being similar. Since K is proper and closed, Dom K = CK × DK is nonempty. ¯ > − for all x ∈ X and for any x¯ ∈ CK , cl1 K x ¯ y ¯ Take y¯ ∈ DK . By definition, cl1 K x y ¯ is a proper lower semicontinuous convex function is finite by Lemma 3.2. Hence, cl1 K · y ¯ and CK ⊂ dom cl1 K · y . ¯ y ¯ = − and the definition of Take y¯ ∈ DK . Then there exists x¯ ∈ X such that cl1 K x ¯ = −. the closure of a convex function implies that cl1 K · y The following result is a simple extension of Lemma 1 in Rockafellar [18]. Lemma 3.4. Let K X × Y → − + be a proper closed saddle function and let F and G be its convex and concave parent, respectively. Then F X × Y ∗ → ∪ + and −G X ∗ × Y → ∪ + are proper lower semicontinuous convex functions, such that


410


dom F ⊂ CK × Y ∗ , dom −G ⊂ X ∗ × DK . Moreover, F is the restriction of the conjugate × Y ∗ , i.e., function −G ∗ X ∗∗ × Y ∗ → ∪ + to the norm closed subspace X F x y ∗ = −G ∗ x ˆ y ∗ and −G is the restriction of the conjugate function F ∗ X ∗ × Y ∗∗ → ∪ + to the norm closed subspace X ∗ × Y, i.e., −G x∗ y = F ∗ x∗ y ˆ Proof. Take a proper closed saddle function K. Then K, cl1 K, and cl2 K have the same convex parent F according to Lemma 3.1. Hence, F is closed and convex because it is a supremum of closed convex functions cl1 K · y + · y on X × Y ∗ . In a similar way one has that −G is closed and convex. The properness and closedness of K imply that the set Dom K = CK × DK is nonempty. The inclusions of the domains hold from Lemma 3.3 and from the closedness of K. For x ∈ CK , the function −cl2 K x · is a proper lower semicontinuous convex function according to Lemma 3.3 and F x · is its conjugate. Hence, F x · also is a proper function (see Aubin and Ekeland [1, p. 201, Theorem 2]). This and the closedness of F imply that F is a proper function. Analogously one obtains that −G is a proper function. Further, since F x · is the conjugate of the closed convex function −cl2 K x · (see (3.3) and Lemma 3.3), from Fenchel duality (2.2) we have F x · ∗ = −cl2 K x · on Y , i.e., −cl2 K x y = sup y y ∗ − F x y ∗ y ∗ ∈Y ∗

For similar reasons,

cl1 K x y = sup G x∗ y + x x∗ x∗ ∈X ∗

(3.5)

As we said above, F is also the convex parent of cl1 K, i.e., F x y ∗ = supcl1 K x y + y y ∗ y∈Y

Combining the latter with (3.5) one obtains F x y ∗ =

sup

x∗ y ∈X ∗ ×Y

G x∗ y + y y ∗ + x x∗ %

(3.6)

hence, F x y ∗ is the restriction of the conjugate function −G ∗ X ∗∗ × Y ∗ → ∪ + × Y ∗ . Similarly one obtains that to the norm closed subspace X − G x∗ y =

sup

xy ∗ ∈X×Y ∗

−F x y ∗ + y y ∗ + x x∗

(3.7)

so −G x∗ y is the restriction of the conjugate function F ∗ X ∗ × Y ∗∗ → ∪ + to the norm closed subspace X ∗ × Y and the proof is then complete. 4. Subdifferential of a saddle function. Partially ball weakly inf-compact saddle functions. Definition and properties. The notion of the subdifferential of a saddle function K X × Y → − + is introduced by Rockafellar as the multivalued mapping ∗ ∗ K X × Y → 2X ×Y defined by K x y = x∗ y ∗ ∈ Z ∗ x∗ is a subgradient of the convex function K · y at x and − y ∗ is a subgradient of the convex function − K x · at y The (possibly empty) set K x y is called the subdifferential of K at x y (see Rockafellar [16, 19]). The domain of K is defined by Dom K = x y ∈ X × Y K x y = . It is clear from the definitions and from (2.5) that when K is proper, Dom K ⊂ Dom K

(4.1)


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The original finite dimensional version of the following lemma is due to Rockafellar and can be found in Rockafellar [18, Lemma 4]. Our proof follows the same steps. Lemma 4.1. For a proper closed saddle function K X × Y → − + the following are equivalent: ˆ ∈ F x y ∗ ; (a) x∗ y (b) x ˆ y ∗ ∈ −G x∗ y ; (c) x∗ −y ∗ ∈ K x y . Any of these conditions implies that the values F x y ∗ , G x∗ y , and K x y are finite. Proof. First, we will show that (a) and (b) are equivalent. Since by Lemma 3.4, F is a proper closed convex function, we have by (2.4) that ˆ ∈ F x y ∗ ⇐⇒ x ˆ y∗ ∈ F ∗ x∗ y ˆ x∗ y ˆ + F x y ∗ = x x∗ + y y ∗ ⇐⇒ F ∗ x∗ y By Lemma 3.4 again, F ∗ x∗ y ˆ = −G x∗ y , hence x∗ y ˆ ∈ F x y ∗ ⇐⇒ −G x∗ y + F x y ∗ = x x∗ + y y ∗ which is

a ⇐⇒ −G x∗ y + F x y ∗ = x x∗ + y y ∗

Since by Lemma 3.4, −G is a proper closed convex function, we have by (2.4) that ∗ x ˆ y ∗ ∈ −G x∗ y ⇐⇒ x∗ y ∈ −G ˆ x ˆ y∗

ˆ y ∗ −G x∗ y = x x∗ + y y ∗ ⇐⇒ −G ∗ x By Lemma 3.4, −G ∗ x ˆ y ∗ = F x y ∗ ; hence x ˆ y ∗ ∈ −G x∗ y ⇐⇒ F x y ∗ −G x∗ y = x x∗ + y y ∗ which is

b ⇐⇒ F x y ∗ − G x∗ y = x x∗ + y y ∗

Finally, (a) and (b) are equivalent, since a ⇐⇒ F x y ∗ − G x∗ y = x x∗ + y y ∗ ⇐⇒ b ∗

(4.2)

∗

and either of those implies the finiteness of F x y and G x y . Next, we will show that (c) implies (a) and (b). When (c) holds it implies that K x y is finite and by the definition of K we have that x∗ ∈ 1 K x y , y ∗ ∈ 2 −K x y , i.e., K z y ≥ K x y + z − x x∗ ∗

∀ z ∈ X

−K x w ≥ −K x y + w − y y

∀w ∈ Y

K x y − x x∗ ≤ K z y − z x∗

∀ z ∈ X

⇐⇒ K x y + y y ∗ ≥ K x w + w y ∗

∀w ∈ Y

⇐⇒ K x y − x x∗ ≤ G x∗ y K x y + y y ∗ ≥ F x y ∗

(4.3)

Adding the last two inequalities we obtain F x y ∗ − G x∗ y ≤ x x∗ + y y ∗ . The opposite inequality being obvious, we have F x y ∗ − G x∗ y = x x∗ + y y ∗ . It follows from (4.2) that (a) and (b) hold.

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Finally, let us suppose that (a) and (b) hold. The function F x · is a proper lower semicontinuous convex function and yˆ ∈ 2 F x y ∗ . Since F x · is the conjugate of −cl2 K x · , it follows from (2.4) that yˆ ∈ 2 F x y ∗ ⇐⇒ y ∗ ∈ 2 −cl2 K x y ⇐⇒ F x y ∗ − y y ∗ = cl2 K x y The function −G · y is a proper lower semicontinuous convex function and xˆ ∈ 1 −G x∗ y . Since −G · y is the conjugate of cl1 K · y , we have from (2.4) xˆ ∈ 1 −G x∗ y ⇐⇒ x∗ ∈ 1 cl1 K x y ⇐⇒ G x∗ y + x x∗ = cl1 K x y From (4.2), we have that the left-hand sides of the last two equalities are equal; hence we obtain that cl1 K x y = cl2 K x y , and that, in particular, they are both equal to K x y . Hence, F x y ∗ = K x y + y y ∗ G x∗ y = K x y − x x∗ which entails (4.3). However, it was already shown that (4.3) is equivalent to (c). The proof is then complete. Combining Lemma 3.1 and Lemma 4.1, one easily obtains that if K is a proper closed saddle function and L is a saddle function equivalent to K, i.e., L ∼ K, then K = L. Hence, the subdifferential of a proper closed saddle function K depends only on the equivalence class to which K belongs and does not depend on its representatives. It is established by Rockafellar that the domain of the subdifferential of a proper closed saddle function K X × Y → − + is nonempty when one of the spaces X and Y is reflexive (see Rockafellar [20]). We will extend this property (see Theorem 4.1) for a class of closed saddle functions defined on product of Banach spaces introduced by the following Definition 4.1. Let X Y be Banach spaces and let K X × Y → − + be a saddle function. We say that K is X-bwc if for some y0 ∈ DK the function cl1 K · y0 is bwc. Respectively, we say that K is Y -bwc if for some x0 ∈ CK the function −cl2 K x0 · is bwc. The function K is said to be partially ball weakly inf-compact (pbwc for short) if it is X-bwc or Y -bwc. When for any x0 y0 ∈ Dom K it holds that cl1 K · y0 or −cl2 K x0 · is bwc, then K is said to be totally partially ball weakly inf-compact (tpbwc for short). Let us note that when one of the spaces X and Y , say X, is reflexive, then any saddle function K on X × Y is tpbwc because cl1 K · y is X-bwc for all y ∈ Y , which is ensured by the weak compactness of the closed unit ball BX . From the very definition it is clear that K is pbwc whenever it is tpbwc, but when both spaces are not reflexive there exist pbwc saddle functions which are not tpbwc as we can see from the following Example. Let X and Y be two Banach spaces which are nonreflexive. Fix any function g X → ∪+ that is convex, lower semicontinuous and bwc with g 0 = 0 and g x > 0 for all x ∈ X\0, and such that for C = dom g the set BX ∩ cl C is not w X X ∗ compact and is nonempty. Define a function K from X × Y into − + by  1 − y g x if x ∈ C y ∈ BY    if x ∈ C y ∈ BY K x y = −    + otherwise.


413

Since cl2 K x · = K x · for all x ∈ X and since    K · y if y < 1  cl1 K · y = *cl C · if y = 1    − if y > 1 (here, for a subset S, *S denotes the indicator function, i.e, *S x = 0 if x ∈ S and *S x = + otherwise) it is not difficult to check that K is a proper closed saddle function with Dom K = C × BY . As for y0 = 0 ∈ BY one has cl1 K · y0 = K · y0 = g · which is bwc, we get that the function K is pbwc. We claim that K is not tpbwc. Fix any x0 y0 ∈ Dom K = C × BY with y0 = 1. On the one hand, cl1 K · y0 = *cl C · which is not bwc since BX ∩ cl C is not w X X ∗ compact. On the other hand, 1 − y g x0 if y ∈ BY cl2 K x0 y = − if y ∈ BY % hence for r = 0, BY ∩ y ∈ Y − cl2 K x0 y ≤ r = BY ∩ BY = BY which is not w Y Y ∗ compact. So the claim is established. An example of such a function g X → ∪ + with the properties listed above is given in X = l1 by g x =

n=1

2n xn

for all x ∈ l1 x = xn n=1

Indeed, one has g ≤ r = for r < 0, g ≤ 0 = 0 for r = 0, and g ≤ r ⊂ −a a with n n a = r/2n n=1 and −a a = x ∈ l1 − r/2 ≤ xn ≤ r/2 ∀ n is easily seen to be totally bounded and closed and hence · compact. So, the function g is a convex, lower semicontinuous bwc function with g 0 = 0 and g x > 0 for all x ∈ X\0. Concerning the above properties required for g, it remains to show that L = BX ∩ cl C is not w X X ∗ compact. Let ek with k ≥ 1 be the standard basis of l1 = X and c = ∗ k cn n=1 ∈ l = X with 0 < cn < cn+1 and cn → 1 so c = 1. Since e ∈ L and k c e = ck → 1 one has supx∈L c x = 1. On the other hand, for each x ∈ L\0, c x ≤

n=1

cn xn
0 and consider the function K0 as above with C = x0 + 2BX and D = y0 + 2BY . Observe that (4.5) entails that DK0 ⊂ DK and that, for any y ∈ DK0 we have by (4.4) that K · y ≤ K0 · y

and hence

cl1 K · y ≤ cl1 K0 · y

So cl1 K0 · y ≤ r ⊂ cl1 K · y ≤ r for all r ∈ and y ∈ DK0 . This says in particular that cl1 K0 · y0 is bwc; hence K0 is pbwc. Then Theorem 4.1 ensures that Dom K0 = and hence (4.6) gives some x y ∈ C × D such that K x y = . The proof is then complete. Another interesting property of a closed proper pbwc saddle function is that the graphs of the subdifferentials of its parent functions are completely determined by the graph of its subdifferential. Lemma 4.3. Let X Y be Banach spaces and let K X × Y → − + be a proper closed pbwc saddle function. Then gph F and gph −G are completely determined by the elements of gph K. Proof. Suppose that K is X-bwc. Lemma 4.1 shows that x ˆ y ∗ ∈ −G x∗ y exactly when x∗ −y ∗ ∈ K x y , so by Lemma 4.2(b), gph −G is completely determined by gph K. From Lemma 3.4 we know that F is a proper closed convex function. The density of dom F in the nonempty set dom F implies that dom F is nonempty. Let x∗ y ∗∗ ∈ F x y ∗ . From Lemma 4.2(c) we have that x∗ y ∗∗ ∈ F x y ∗ exactly when x∗ y ∗∗ ∈ ˆ y ∗ . Set W to be the Banach space on which the concave parent G of K is −G ∗ x defined, i.e., W = X ∗ × Y . For the conjugate function −G ∗ , Lemma 4.2(a) gives that × Y ∗ . This combined with Theorem 2.1 ensures that: dom −G ∗ ⊂ X  x∗ y ∗∗ ∈ −G ∗ x ˆ y ∗ ⇐⇒ there exists a net x y ∗ ∈A ∈ W ∗    converging to x ˆ y ∗ in the norm topology of W ∗ and a bounded net (4.7) ∗ converging to x∗ y ∗∗ in the w W ∗∗ W ∗ topology  x y ∈A ∈ W   such that x y∗ ∈ −G x∗ y ∀ ∈ A Lemma 4.1 implies that x y∗ ∈ −G x∗ y exactly when x∗ −y∗ ∈ K x y . Hence, (4.7) may be rewritten as:  ∗ ∗∗ x y ∈ F x y ∗ ⇐⇒ there exists a net x y ∗ ∈A ∈ W ∗    converging to x ˆ y ∗ in the norm topology of W ∗ and a bounded net (4.8) converging to x∗ y ∗∗ in the w W ∗∗ W ∗ topology  x∗ y ∈A ∈ W   ∗ ∗ such that x −y ∈ K x y ∀ ∈ A The latter says that gph F is completely determined by the elements of gph K. With any saddle function K X × Y → − + one may associate a monotone (see ∗ ∗ below) multivalued operator TK X × Y → 2X ×Y given by TK x y = x∗ y ∗ ∈ X ∗ × Y ∗ x∗ −y ∗ ∈ K x y When K is proper and closed, we saw in Lemma 4.1 that K depends only on the equivalence class. The above definition ensures that in this case the same holds for the operator TK . This operator was introduced in relation with minimax problems by Rockafellar [20] who

416


proved that in the case when one of the Banach spaces involved is assumed to be reflexive, TK is maximal monotone whenever K is proper and closed (see Rockafellar [20, Theorem 3]). We will show that this is still true when K is a proper closed partially ball weakly inf-compact saddle function defined on a product Banach space. Closed relationship between proper closed saddle functions on X × X and maximal monotone operators on X is established by Krauss [8]. Let us recall that a set-valued mapping S from a Banach space X to its dual X ∗ is said to be monotone if x∗ ∈ S x , y ∗ ∈ S y imply x − y x∗ − y ∗ ≥ 0. The operator S is said to be maximal monotone if S is monotone and S has no proper monotone extension, i.e., if x x∗ ∈ X × X ∗ is such that the monotone relation x − y x∗ − y ∗ ≥ 0 holds for all y y ∗ ∈ gph S, then x x∗ ∈ gph S (cf., e.g., Phelps [15], Rockafellar [21], and Simons [23]). Theorem 4.3. Let X Y be Banach spaces and let K X × Y → − + be a closed proper pbwc saddle function. Then the operator TK is maximal monotone. Proof. Let K be X-bwc. By Lemma 4.2(b) we know that the range of −G lies × Y ∗ . By Lemma 4.1 and by the definition of TK it is clear that x∗ y ∗ ∈ TK x y in X exactly when x ˆ y ∗ ∈ −G x∗ y . The latter, being the subdifferential of a proper closed convex function, is maximal monotone (see Rockafellar [21]). This implies the maximal monotonicity of TK . Let us note that in the paper of Pak [14] one can find the statement of the above result for an arbitrary proper closed saddle function defined on a product of arbitrary Banach spaces, but the proof presented there implicitly presumes reflexivity. 5. Integrability of the subdifferential of a proper closed partially ball weakly inf-compact saddle function. The integration of subdifferentials concerns the problem whether or not the condition that the subdifferental of g contains the subdifferental of f implies that g and f differ by a constant. The famous Rockafellar integration result (see Rockafellar [21]) states that the inclusion f x ⊂ g x

for all x ∈ X

entails that g and f are equal up to a constant whenever f g X → ∪ + are proper, lower semicontinuous convex functions and X is a Banach space. Here we are interested in the integrability of the subdifferential of a saddle function on the product of Banach spaces. The result is established for Lipschitz saddle functions by Correa and Thibault [4], some generalizations for directionally Lipschitz saddle functions can be found in Thibault and Zlateva [25]. We consider two proper closed saddle functions K L X × Y → − + defined on a product Banach space and we are interested whether the pbwc condition on one of the functions K and L and the inclusion L ⊂ K entail that K and L are equivalent up to a finite additive constant. First we will consider the case when the inside for the subdifferential inclusion function L is supposed to be pbwc. Theorem 5.1. Let X Y be Banach spaces and let L X × Y → − + be a proper closed pbwc saddle function. Then for any proper closed saddle function K X × Y → − + the condition L x y ⊂ K x y

∀ x y ∈ Dom L

implies that K and L are equivalent up to a finite additive constant c, i.e., K ∼ L + c .


417

Proof. Let K and L be saddle functions satisfying the assumptions of Theorem 5.1. Let us denote by F and F the convex parents of L and K, respectively and by G and G the concave parents of L and K, respectively. Since L and K are proper and closed we know by Lemma 3.4 that the functions F F −G , and −G are proper and lower semicontinuous convex functions. Since L is supposed to be pbwc, without loss of generality we suppose that L is X-bwc. First, we will show that (5.1) −G ⊂ −G ∗ × Y . Take any x ˆ y∗ ∈ Since L is X-bwc, by Lemma 4.2(b) we have Rge −G ⊂ X ∗ ∗ ∗ ∗ ∗ −G x y . Lemma 4.1 for L ensures that x −y ∈ L x y , so x −y ∈ K x y . Lemma 4.1 for K yields that x ˆ y ∗ ∈ −G x∗ y . Hence, we have proved (5.1). Second, we will show that (5.2) F ⊂ F Take any x∗ y ∗∗ ∈ F x y ∗ . Since L is X-bwc, Lemma 4.2(c) gives that the latter ˆ y ∗ . Since −G ∗ is the conjugate function of the lower implies x∗ y ∗∗ ∈ −G ∗ x semicontinuous convex function −G , then Theorem 2.1 applied to it gives that (recalling the notation W = X ∗ × Y )  x∗ y ∗∗ ∈ −G ∗ x ˆ y ∗ ⇒ there exists a net x∗∗ y ∗ ∈A ∈ W ∗    converging to x ˆ y ∗ in the norm topology of W ∗ and a bounded net (5.3) converging to x∗ y ∗∗ in the w W ∗∗ W ∗ topology  x∗ y ∈A ∈ W   such that x∗∗ y∗ ∈ −G x∗ y ∀ ∈ A The X-bwc property of L yields, according to Lemma 4.2(b), that x∗∗ = x for some x ∈ X. From Lemma 4.1 x y∗ ∈ −G x∗ y implies that x∗ −y∗ ∈ L x y . From L ⊂ K we have x∗ −y∗ ∈ K x y which, by Lemma 4.1, yields that x y∗ ∈ −G x∗ y . Since x y∗ ∈ −G x∗ y for all ∈ A and since the net x y∗ ∈A ∈ W ∗ con verges to x ˆ y ∗ in the norm topology of W ∗ and the bounded net x∗ y ∈A ∈ W converges to x∗ y ∗∗ in the w W ∗∗ W ∗ topology, Theorem 2.1 implies that x∗ y ∗∗ ∈ ˆ y ∗ . As F u v∗ = −G ∗ u ˆ v∗ for all u v∗ ∈ X × Y ∗ by Lemma 3.4, we −G ∗ x ∗ ∗∗ ∗ deduce that x y ∈ F x y . Hence, (5.2) is established. The functions F and F are both proper lower semicontinuous convex functions on the Banach space X × Y ∗ satisfying F ⊂ F . Hence, we can apply Rockafellar [21, Theorem B] (see also Thibault and Zagrodny [24, Corollary 2.2]) to deduce that there exists a finite constant c such that F x y ∗ = F x y ∗ + c

∀ x y ∗ ∈ X × Y ∗

(5.4)

The functions −G and −G are both proper lower semicontinuous convex functions satisfying −G ⊂ −G . Analogous reasoning gives a finite constant d such that G x∗ y = G x∗ y + d

∀ x∗ y ∈ X ∗ × Y

(5.5)

Now using (3.6) from Lemma 3.4, (5.5), and (5.4) we obtain that F x y ∗ = =

sup

G x∗ y + y y ∗ + x x∗

sup

G x∗ y + y y ∗ + x x∗ + d

x∗ y ∈X ∗ ×Y x∗ y ∈X ∗ ×Y

= F x y ∗ + d = F x y ∗ − c + d and c = d because F is finite at some point. From (5.4) and (5.5) it is clear that the functions K and L + c have the same parent functions. By Lemma 3.1 one obtains that K ∼ L + c and K and L are equivalent up to the additive constant c. The proof is then complete.


418


Second we consider the case when the outside for the subdifferential inclusion function K is supposed to be pbwc. Theorem 5.2. Let X Y be Banach spaces and let K X × Y → − + be a proper closed pbwc saddle function. Then for any proper closed saddle function L X × Y → − + the conditions 1 cl1 L ·y ⊂ 1 K ·y

and

2 −cl2 L x· ⊂ 2 −K x·

∀ xy ∈ X ×Y

(5.6)

imply that K and L are equivalent up to a finite additive constant c, i.e., K ∼ L + c . Proof. Let x ¯ y ¯ ∈ Dom L. By Lemma 3.3 we know that the function cl1 L · y ¯ is a ¯ By the density proper lower semicontinuous convex function and that x¯ ∈ dom cl1 L · y . of the domain of the subdifferential of a closed convex function in its effective domain ¯ ⊂ dom 1 K · y , ¯ the latter by (see (2.3)) it follows that there exists x˜ ∈ dom 1 cl1 L · y ¯ Hence, cl1 K · y ¯ is a proper closed (5.6). From (2.5) we have that x˜ ∈ dom cl1 K · y . convex function also and further the first inclusion of (5.6) yields 1 cl1 L · y ¯ ⊂ 1 cl1 K · y . ¯ We can then apply Rockafellar [21, Theorem B] (see also Thibault and Zagrodny [24, Corollary 2.2]) to obtain that there exists a finite constant c y ¯ such that ¯ = cl1 K x y ¯ + c y ¯ ∀x ∈ X cl1 L x y

and

¯ ≡ dom cl1 K · y ¯ dom cl1 L · y

(5.7)

Analogously, one obtains that there exists a finite constant d x ¯ such that cl2 L x ¯ y = cl2 K x ¯ y + d x ¯ ∀y ∈ Y

and

¯ · ≡ dom −cl2 K x ¯ · dom −cl2 L x

(5.8)

Recall that (5.7) and (5.8) hold for all x ¯ y ¯ ∈ Dom L. From (5.7), (5.8), and Lemma 3.2, it follows that x ¯ y ¯ ∈ Dom K, so Dom L ⊂ Dom K. To establish the opposite inclusion of domains, let us take any x ¯ y ¯ ∈ Dom K. Sup¯ = − and 1 cl1 L x y ¯ ≡ X ∗ for any x ∈ X. The pose that y¯ ∈ DL . Then cl1 L · y first inclusion in (5.6) ensures that K · y ¯ is everywhere subdifferentiable; hence from ¯ and 1 cl1 K · y ¯ = 1 K · y ¯ = X ∗ , which yields a contradiction, (2.5), K · y ¯ = cl1 K · y , ¯ is a proper closed convex function. By similar argument we obtain that because cl1 K · y x¯ ∈ CL ; hence Dom K ⊂ Dom L and, finally, Dom L = Dom K. As K is pbwc, we may suppose that K is X-bwc. Take y0 ∈ DK = DL such that cl1 K · y0 is bwc. Fix r ∈ and n ∈ and consider the sublevel set Pr n = cl1 L · y0 ≤ r ∩ nBX . Since the function cl1 L · y0 is convex and lower semicontinuous, the latter set is weakly closed. By (5.7), Pr n ⊂ cl1 K · y0 ≤ r − c y0 ∩ nBX , which is weakly compact. Hence, Pr n is weakly compact, which implies that the closed saddle function L is X-bwc also. Then by Theorem 4.1, Dom L = . Take any x∗ y ∗ ∈ L x y . In particular, x∗ ∈ 1 L · y x . From (2.5) and the properness of L · y , we have that L x y = cl1 L x y and that 1 cl1 L · y x = 1 L · y x . The latter says that x∗ ∈ 1 cl1 L · y x . The assumption (5.6) gives that x∗ ∈ 1 K · y x . Analogously one obtains that −y ∗ ∈ 2 −K x ·

y . Hence, x∗ y ∗ ∈ K x y . The arbitrariness of x∗ y ∗ ensures that L x y ⊂ K x y for all x y ∈ Dom L. The assumptions of Theorem 5.1 being satisfied, we conclude that the result is established. Acknowledgments. The authors are grateful for the helpful comments made by a referee. The authors express their gratitude to Dr. Milen Ivanov from Sofia University for suggesting the example. The research of the second author was supported by a Marie Curie fellowship of the European Community program, Improving Human Potential under Contract HPMF-CT-2001-01345.


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