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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 oper
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