A Simple Proof of the Baillon-Haddad Theorem on Open Subsets of Hilbert Spaces

Daniel Wachsmuth; Gerd Wachsmuth

doi:10.68381/jca30059

Abstract

We give a simple proof of the Baillon-Haddad theorem for convex functions defined on open and convex subsets of Hilbert spaces. We also state some generalizations and limitations. In particular, we discuss equivalent characterizations of the Lipschitz continuity of the derivative of convex functions on open and convex subsets of Banach spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Daniel Wachsmuth

Institut für Mathematik, Universität Würzburg, Germany

daniel.wachsmuth@mathematik.uni-wuerzburg.de

Gerd Wachsmuth

Institut für Mathematik, Brandenburgische Technische Universität, Cottbus-Senftenberg, Germany

gerd.wachsmuth@b-tu.de

Keywords

Baillon-Haddad theorem
cocoercivity
strong smoothness

Mathematics Subject Classification

26B25, 47H05, 47N10, 49J50

References

J.-B. Baillon,G. Haddad: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones, Israel J. Math. 26/2 (1977) 137–150.https://doi.org/10.1007/BF03007664 Google Scholar
H. H. Bauschke, P. L. Combettes: The Baillon-Haddad Theorem revisited, J. Convex Analysis 17/3-4 (2009) 781–787.https://doi.org/10.68381/jca17051 Google Scholar
H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-9467-7 Google Scholar
Y. Drori: On the properties of convex functions over open sets, J. Convex Analysis 27/4 (2020) 1303–1314.https://doi.org/10.68381/jca27069 Google Scholar
V. Kryštof, L. Zajíček: Functions on a convex set which are both ω-semiconvex and ω-semiconcave, J. Convex Analysis 29/3 (2022) 837–856.https://doi.org/10.68381/jca29049 Google Scholar
P. Pérez-Aros, E. Vilches: An enhanced Baillon-Haddad theorem for convex functions defined on convex sets, Appl. Math. Optimization 83/3 (2019) 2241–2252.https://doi.org/10.1007/s00245-019-09626-6 Google Scholar
C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.

[1] J.-B. Baillon,G. Haddad: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones, Israel J. Math. 26/2 (1977) 137–150.https://doi.org/10.1007/BF03007664 Google Scholar

[2] H. H. Bauschke, P. L. Combettes: The Baillon-Haddad Theorem revisited, J. Convex Analysis 17/3-4 (2009) 781–787.https://doi.org/10.68381/jca17051 Google Scholar

[3] H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-9467-7 Google Scholar

[4] Y. Drori: On the properties of convex functions over open sets, J. Convex Analysis 27/4 (2020) 1303–1314.https://doi.org/10.68381/jca27069 Google Scholar

[5] V. Kryštof, L. Zajíček: Functions on a convex set which are both ω-semiconvex and ω-semiconcave, J. Convex Analysis 29/3 (2022) 837–856.https://doi.org/10.68381/jca29049 Google Scholar

[6] P. Pérez-Aros, E. Vilches: An enhanced Baillon-Haddad theorem for convex functions defined on convex sets, Appl. Math. Optimization 83/3 (2019) 2241–2252.https://doi.org/10.1007/s00245-019-09626-6 Google Scholar

[7] C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021 Google Scholar