Math. Program., Ser. B (2018) 168:433–448
FULL LENGTH PAPER
Representative functions of maximally monotone
operators and bifunctions
· Nicolas Hadjisavvas
Received: 4 August 2015 / Accepted: 22 April 2016 / Published online: 11 May 2016
© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016
Abstract The aim of this paper is to show that every representative function of a max-
imally monotone operator is the Fitzpatrick transform of a bifunction corresponding
to the operator. In fact, for each representative function ϕ of the operator, there is a
family of equivalent saddle functions (i.e., bifunctions which are concave in the ﬁrst
and convex in the second argument) each of which has ϕ as Fitzpatrick transform.
In the special case where ϕ is the Fitzpatrick function of the operator, the family of
equivalent saddle functions is explicitly constructed. In this way we exhibit the relation
between the recent theory of representative functions, and the much older theory of
saddle functions initiated by Rockafellar.
Keywords Maximal monotonicity · Fitzpatrick function · representative function ·
Mathematics Subject Classiﬁcation 47H05 · 47H04 · 49J53 · 90C33
Part of this work was done when Nicolas Hadjisavvas was visiting the Università Cattolica del Sacro
Cuore, and the Università degli Studi di Milano–Bicocca, Italy. The author wishes to thank the
Universities for their hospitality. Nicolas Hadjisavvas was supported by the startup research Grant No.
SR141001 of the King Fahd University of Petroleum and Minerals.
Università Cattolica del Sacro Cuore, Milan, Italy
King Fahd University of Petroleum and Minerals, Dhahran, Kingdom of Saudi Arabia
Università degli Studi di Milano–Bicocca, Milan, Italy