Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative

Abstract

Several deeper results on maximal monotone operators have recently found simpler proofs using Fitzpatrick functions. In this paper, we study a sequence of Fitzpatrick functions associated with a monotone operator. The first term of this sequence coincides with the original Fitzpatrick function, and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties. It is shown that for any maximal cyclically monotone operator, the pointwise supremum of the sequence of Fitzpatrick functions is closely related to Rockafellar’s antiderivative. Several examples are explicitly computed for the purpose of illustration. In contrast to Rockafellar’s result, a maximal 3-cyclically monotone operator need not be maximal monotone. A simplified proof of Asplund’s observation that the rotation in the Euclidean plane by π/n is n-cyclically monotone but not (n+1)-cyclically monotone is provided. The Fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail. We conclude with a new proof of Moreau’s result concerning the convexity of the set of proximal mappings.

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Fitzpatrick functions, cyclic monotonicity
and Rockafellar’s antiderivative
Sedi Bartz∗, Heinz H. Bauschke†, Jonathan M. Borwein‡, Simeon Reich§, and Xianfu Wang¶
December 31, 2005 — Version 1.23
Abstract
Several deeper results on maximal monotone operators have recently found simpler proofs using
Fitzpatrick functions. In this paper, we study a sequence of Fitzpatrick functions associated
with a monotone operator. The first term of this sequence coincides with the original Fitzpatrick
function, and the other terms turn out to be useful for the identification and characterization of
cyclic monotonicity properties. It is shown that for any maximal cyclically monotone operator,
the pointwise supremum of the sequence of Fitzpatrick functions is closely related to Rockafellar’s
antiderivative. Several examples are explicitly computed for the purpose of illustration. In
contrast to Rockafellar’s result, a maximal 3-cyclically monotone operator need not be maximal
monotone. A simplified proof of Asplund’s observation that the rotation in the Euclidean plane
by π/n is n-cyclically monotone but not (n+1)-cyclically monotone is provided. The Fitzpatrick
family of the subdifferential operator of a sublinear and of an indicator function is studied in
detail. We conclude with a new proof of Moreau’s result concerning the convexity of the set of
proximal mappings.
2000 Mathematics Subject Classification: Primary 47H05; Secondary 52A41, 90C25.
Keywords: Convex function, Fenchel conjugate, Fitzpatrick function, cyclically monotone opera-
tor, maximal monotone operator, subdifferential operator, sublinear function.
∗Department of Mathematics, The Technion—Israel Institute of Technology, 32000 Haifa, Israel.
bartz@techunix.technion.ac.il.
†Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail:
heinz.bauschke@ubc.ca.
‡Faculty of Computer Science, Dalhousie University, 6050 University Avenue, Halifax, Nova Scotia B3H 1W5,
Canada. E-mail: jborwein@cs.dal.ca.
§Department of Mathematics, The Technion—Israel Institute of Technology, 32000 Haifa, Israel.
sreich@techunix.technion.ac.il.
¶Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail:
shawn.wang@ubc.ca.
E-mail:
E-mail:
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1Introduction
Throughout this paper, we assume that
X is a real Banach space, (1)
with norm ?·?, with continuous dual space X∗, and with the pairing p = ?·,·? between X and X∗.
Our aim is to provide new results on monotone operators and their associated Fitzpatrick func-
tions. We denote the graph of a set-valued operator A: X → 2X∗by graA =?(x,x∗) ∈ X × X∗|
Definition 1.1 Let A: X → 2X∗. Then A is n-cyclically monotone if n ∈ {2,3,...} and
(a1,a∗
1) ∈ graA
...
(an,a∗
n) ∈ graA
an+1= a1
x∗∈ Ax?and recall the following notions.








?

⇒
n
?
i=1
?ai+1− ai,a∗
i? ≤ 0.(2)
The operator A is monotone if it is 2-cyclically monotone; equivalently, if
(x,x∗) ∈ graA
(y,y∗) ∈ graA
⇒?x − y,x∗− y∗? ≥ 0.(3)
The operator A is cyclically monotone if for every n ∈ {2,3,...}, A is n-cyclically monotone.
Definition 1.2 Let A: X → 2X∗and let n ∈ {2,3,...}. Then A is maximal n-cyclically monotone
if A is monotone and no proper extension of A is n-cyclically monotone; A is maximal cyclically
monotone if A is cyclically monotone and no proper extension of A is cyclically monotone; A is
maximal monotone if A is maximal 2-cyclically monotone.
Zorn’s Lemma guarantees that every n-cyclically monotone operator admits an extension, in the
sense of enlargement of the graph, to a maximal n-cyclically monotone extension. An analogous
result holds for cyclically monotone operators. Monotone operators play a fundamental role in
optimization; the reader is referred to [14, 42, 43, 48] for further information. Linear maximal n-
cyclically monotone operators are discussed in [1]. One of the most remarkable results in monotone
operator theory is due to Rockafellar [38] who proved that the maximally cyclically monotone oper-
ators are precisely the subdifferential operators of functions that are convex, lower semicontinuous,
and proper. Hence every maximally cyclically monotone operator admits an antiderivative, which
is unique up to an additive constant, see Fact 3.2 below. Other fundamental results concern char-
acterizations of maximal monotonicity and the maximality of the sum of two maximal monotone
operators under a constraint qualification. Through the use of the Fitzpatrick function [21], such
results have recently found dramatically simpler proofs; see, e.g., [9] and [45]. Fitzpatrick functions
play also a key role in various works on monotone operators [12, 16, 17, 18, 28, 29, 34, 44]. Let us
now state the definition of a Fitzpatrick function.
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Definition 1.3 Let A: X → 2X∗. The Fitzpatrick function of A is
FA: X × X∗→ ]−∞,+∞] : (x,x∗) ?→ sup
(a,a∗)∈graA
??x,a∗? + ?a,x∗? − ?a,a∗??. (4)
Fitzpatrick [21] established the following fundamental properties.
Fact 1.4 Let A: X → 2X∗be maximal monotone. Then FA is convex, lower semicontinuous,
proper, p ≤ FA, and p = FAon graA. Moreover, FAis the smallest function with these properties.
In this paper, we provide several novel results concerning Fitzpatrick functions and monotone
operators. These results can be roughly classified into five categories, which also correspond to the
contents of the remaining five sections.
ΠIn Section 2, we introduce an increasing sequence of Fitzpatrick functions of order n, where
n ∈ {2,3,...}. The first term of this sequence coincides with the original Fitzpatrick func-
tions, which is a powerful tool in the study of maximal monotone operators. Analogously, we
show that the Fitzpatrick functions of order n captures precisely (maximal) n-cyclic mono-
tonicity properties. We also provide an example of a maximal 3-cyclically monotone operator
that is not maximal monotone (Example 2.16). This is in striking contrast to Rockafellar’s
characterization of maximal cyclically monotone operators.
 The important case of subdifferential operators is considered in greater detail in Section 3.
We show how the Fitzpatrick functions and Rockafellar’s antiderivative arise from a common
ancestor and prove, using deeper results on subgradients, that the sequence of Fitzpatrick
functions converges to a function that has a close relationship to Rockafellar’s antiderivative
(Theorem 3.5).
Ž Section 4 contains several examples of maximal monotone operators where the Fitzpatrick
functions of all orders are computed in closed form. It is possible that outside the graph all
Fitzpatrick functions are either identical (Example 4.2) or pairwise distinct (Example 4.4).
We also provide a simple, more self-contained proof of Asplund’s result [1] that rotations in
the Euclidean plane by π/n are n-cyclically monotone but not (n + 1)-cyclically monotone
(Example 4.6).
 The Fitzpatrick family of a given maximal monotone operator A: X → 2X∗consists of
all convex, lower semicontinuous, and proper functions that are greater than or equal to p
everywhere and equal to p on the graph of A. By Fact 1.4, this family contains FA. In
Section 5, we extend results of Penot [34] and of Burachik and Fitzpatrick [16] by showing
that the Fitzpatrick family is a singleton when A is the subdifferential operator of either a
sublinear function (Theorem 5.3) or of an indicator function (Corollary 5.9).
 The last Section 6 deals with convexity properties of the set of proximal mappings (resol-
vents of subdifferential operators). Moreau showed that in Hilbert space the set of proximal
mappings is convex [33]. We provide a different proof of this result (Theorem 6.7) and also
an example illustrating the nonconvexity of the set of firmly nonexpansive mappings outside
Hilbert space (Example 6.4 and Remark 6.5).
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Notation utilized is standard in Convex Analysis and Monotone Operator Theory; see, e.g., [12],
[43], and [48].
2Fitzpatrick functions and cyclic monotonicity
Let us introduce the following functions which may be interpreted as common ancestors of the
Fitzpatrick function and Rockafellar’s antiderivative.
Definition 2.1 Let A: X → 2X∗, let (a1,a∗
set CA,2,(a1,a∗
n ∈ {3,4,...}. Then the value of the function CA,n,(a1,a∗
is defined by
1) ∈ graA, and let n ∈ {2,3,...}. If n = 2, we
1? + ?a1,x∗? − ?a1,a∗
1): X×X∗→ ]−∞,+∞] at (x,x∗) ∈ X×X∗
1): X × X∗→ ]−∞,+∞] : (x,x∗) ?→ ?x,a∗
1?. Now suppose that
sup
(a2,a∗
2)∈graA
...
n−1)∈graA
(an−1,a∗
?n−2
i=1
?
?ai+1− ai,a∗
i?
?
+ ?x − an−1,a∗
n−1? + ?a1,x∗?;(5)
equivalently, by
sup
(a2,a∗
2)∈graA,
...
n−1)∈graA
(an−1,a∗
?x,x∗? +
?n−2
i=1
?
?ai+1− ai,a∗
i?
?
+ ?x − an−1,a∗
n−1? + ?a1− x,x∗?.(6)
Definition 2.2 (Fitzpatrick functions) Let A: X → 2X∗. For every n ∈ {2,3,...}, the Fitz-
patrick function of A of order n is
FA,n=sup
(a,a∗)∈graA
CA,n,(a,a∗).(7)
The Fitzpatrick function of A of infinite order is FA,∞= supn∈{2,3,...}FA,n.
The first result is immediate from the definition.
Proposition 2.3 Let A: X → 2X∗and let n ∈ {2,3,...}. Then FA,n: X × X∗→ [−∞,+∞] is
convex and lower semicontinuous. At (x,x∗) ∈ X × X∗, the value of FA,nis given by
?n−2
i=1
sup
(a1,a∗
1)∈graA,
...
n−1)∈graA
(an−1,a∗
?x,x∗? +
?
?ai+1− ai,a∗
i?
?
+ ?x − an−1,a∗
n−1? + ?a1− x,x∗?.(8)
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Moreover,
FA,n≥ ?·,·? on graA.(9)
If n = 2, then (8) simplifies to sup(a,a∗)∈graA?x,a∗? + ?a,x∗? − ?a,a∗?, which is the original def-
inition (see Definition 1.3) of the Fitzpatrick function [21] of A at (x,x∗) ∈ X × X∗. Note that
(FA,n)n∈{2,3,...}is a sequence of increasing functions and that FA,n→ FA,∞pointwise. We now
provide a characterization of n-cyclically monotone operators by Fitzpatrick functions of order n
which directly generalizes [28, Proposition 2].
Proposition 2.4 Let A: X → 2X∗and let n ∈ {2,3,...}. Then the following are equivalent.
(i) A is n-cyclically monotone.
(ii) FA,n≤ ?·,·? on graA.
(iii) FA,n= ?·,·? on graA.
Proof. Take (x,x∗) ∈ graA and take n − 1 additional points (a1,a∗
“(i)⇒(ii)”: In view of Definition 1.1,
and thus ?x,x∗? +??n−2
“(ii)⇒(iii)”:
?x − an−1,a∗
?a1− x,x∗? ≤ 0, which implies that A is n-cyclically monotone.
Corollary 2.5 Let A: X → 2X∗. Then A is cyclically monotone ⇔ FA,∞ ≤ ?·,·? on graA ⇔
FA,∞= ?·,·? on graA.
1),...,(an−1,a∗
n−1? + ?a1− x,x∗? ≤ 0
n−1? + ?a1− x,x∗? ≤ ?x,x∗?. Recalling (8)
n−1) in graA, we see that FA,n(x,x∗) ≤ ?x,x∗?.
“(iii)⇒(i)”:
n−1? + ?a1− x,x∗? ≤ ?x,x∗?.
n−1) in graA.
??n−2
1),...,(an−1,a∗
i=1?ai+1− ai,a∗
i??+ ?x − an−1,a∗
i=1?ai+1− ai,a∗
i??+ ?x − an−1,a∗
and taking the supremum over (a1,a∗
This follows from (9).By (8), ?x,x∗? +
??n−2
??n−2
i=1?ai+1− ai,a∗
+ ?x − an−1,a∗
i??
+
Hence
i=1?ai+1− ai,a∗
i??
n−1? +
?
Proof. This is clear from Proposition 2.4 and the fact that FA,∞= supn∈{2,3,...}FA,n.
?
Remark 2.6 For any function f : X × X∗→ ]−∞,+∞], set, as in [21, Definition 2.1], Gf: X →
2X∗: x ?→
monotone, the operator GFA,2is a monotone extension of A (and hence A = GFA,2whenever A is
maximal monotone); see [21, Corollary 3.5]. The following generalization holds. Let A : X → 2X∗
be n-cyclically monotone for some n ∈ {2,3,...}. Then GFA,nis a monotone extension of A.
Consequently, if A is maximal monotone, then A = GFA,n. Indeed, it follows from [21, Lemma 3.3]
that GFA,2is a monotone extension of A. Take (x,x∗) ∈ graA. Then (x∗,x) ∈ ∂FA,2(x,x∗) and
thus, for any (y,y∗) ∈ X × X∗,
FA,2(x + y,x∗+ y∗) − FA,2(x,x∗) ≥ ?y,x∗? + ?x∗,x?.
On the other hand, we have FA,n ≥ FA,2 and also, by Proposition 2.4, FA,n(x,x∗) = ?x,x∗? =
FA,2(x,x∗). Altogether, FA,n(x + y,x∗+ y∗) − FA,n(x,x∗) ≥ ?y,x∗? + ?x∗,x?, and hence (x∗,x) ∈
∂FA,n(x,x∗), i.e., (x,x∗) ∈ graGFA,n.
?x∗∈ X∗| (x∗,x) ∈ ∂f(x,x∗)?. Fitzpatrick proved that for any A: X → 2X∗that is
(10)
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