Abstract

In 1988, Simon Fitzpatrick defined a new convex function FAF_A – nowadays called the Fitzpatrick function – associated with a monotone operator A, and similarly a monotone operator GfG_f associated with a convex function f. This paper deals with two different aspects of Fitzpatrick functions. In the first half, we consider the Fitzpatrick function of the subdifferential of a proper, lower semicontinuous, and convex function. A refinement of the classical Fenchel-Young inequality is derived and conditions for equality are investigated. The results are illustrated by several examples. In the second half, we study the problem, originally posed by Fitzpatrick, of determining when A=GFAA = G_{F_A}. Fitzpatrick proved that this identity is satisfied whenever A is maximal monotone; however, he also observed that it can hold even in the absence of maximal monotonicity. We propose a new condition sufficient for this identity, formulated in terms of the polarity notions introduced recently by Martínez-Legaz and Svaiter. Moreover, on the real line, this condition is also necessary and it corresponds to the connectedness of A.

Author information

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

Heinz H. Bauschke

Department of Mathematics, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada

hbauschk@uoguelph.ca

D. Alexander McLaren

Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

amclaren@uoguelph.ca

Hristo S. Sendov

Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

hssendov@uoguelph.ca

Keywords

  • Convex function
  • Fenchel conjugate
  • Fenchel-Young inequality
  • Fitzpatrick function
  • monotone operator
  • monotone set

Mathematics Subject Classification

26B25, 47H05; 47H04, 52A41, 90C25

References

  1. H. H. Bauschke: Projection Algorithms and Monotone Operators, Ph.D. Thesis, Simon Fraser University, Burnaby, Canada, August 1996; available at http://www.cecm.sfu.ca/preprints/1996pp.html.Google Scholar
  2. H. H. Bauschke, J. M. Borwein: Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators, Pac. J. Math. 189 (1999) 1–20.https://doi.org/10.2140/pjm.1999.189.1Google Scholar
  3. R. S. Burachik, B.. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002) 297–316.https://doi.org/10.1023/A:1020639314056Google Scholar
  4. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth: On the Lambert W function, Adv. Comput. Math. 5 (1996) 329–359.https://doi.org/10.1007/BF02124750Google Scholar
  5. A. L. Dontchev, T. Zolezzi: Well-Posed Optimization Problems, Lecture Notes in Mathematics 1543, Springer, Berlin (1993).https://doi.org/10.1007/BFb0084195Google Scholar
  6. S. Fitzpatrick: Representing monotone operators by convex functions, in: Functional Analysis and Optimization, Workshop / Miniconference (Canberra 1988), Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, Australian National University, Canberra (1988) 59–65.Google Scholar
  7. C. W. Groetsch: Generalized Inverses of Linear Operators, Monographs and Textbooks in Pure and Applied Mathematics 37, Marcel Dekker, New York (1977).Google Scholar
  8. J.-E. Martínez-Legaz, B. F. Svaiter: Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13 (2005) 21–46.https://doi.org/10.1007/s11228-004-4170-4Google Scholar
  9. J.-E. Martínez-Legaz, M. Théra: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001) 243–247.Google Scholar
  10. D. A. McLaren: Notes on the Fitzpatrick Function, M. Sc. Thesis, University of Guelph, Ontario, Canada, August 2005.Google Scholar
  11. J.-J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. Fr. 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  12. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pac. J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  13. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  14. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  15. S. Simons: Subdifferentials are locally maximal monotone, Bull. Aust. Math. Soc. 47 (1993) 465–471.https://doi.org/10.1017/S000497270001529XGoogle Scholar
  16. S. Simons: Minimax and Monotonicity, Lecture Notes in Mathematics 1693, Springer, Berlin (1998).https://doi.org/10.1007/BFb0093633Google Scholar
  17. S. Simons: The Fitzpatrick function and the range of a sum, preprint (2005).Google Scholar
  18. S. Simons, C. Zălinescu: A new proof for Rockafellar’s characterization of maximal monotone operators, Proc. Amer. Math. Soc. 132 (2004) 2969–2972.https://doi.org/10.1090/S0002-9939-04-07462-3Google Scholar
  19. S. Simons, C. Zălinescu: Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005) 1–22.Google Scholar
  20. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

H. H. Bauschke, D. A. McLaren, H. S. Sendov. “Fitzpatrick Functions: Inequalities, Examples, and Remarks on a Problem by S. Fitzpatrick.” Journal of Convex Analysis 13 (2006), No. 3/4, 499–523. https://doi.org/10.68381/jca13043

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.