Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The kernel average for two convex functions and its application to the extension and representation of monotone operators


Authors: Heinz H. Bauschke and Xianfu Wang
Journal: Trans. Amer. Math. Soc. 361 (2009), 5947-5965
MSC (2000): Primary 52A41, 47N10; Secondary 47H05
DOI: https://doi.org/10.1090/S0002-9947-09-04698-4
Published electronically: April 17, 2009
MathSciNet review: 2529920
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide and analyze an average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangian) which can be used for finding an explicit maximal monotone extension of the given monotone operator. This completely settles one of the open problems posed by Fitzpatrick in the setting of reflexive Banach spaces.


References [Enhancements On Off] (What's this?)

  • 1. S. Bartz, H. H. Bauschke, J. M. Borwein, S. Reich, and X. Wang, ``Fitzpatrick functions, cyclic monotonicity, and Rockafellar's antiderivative'', Nonlinear Analysis, vol. 66, pp. 1198-1223, 2007. MR 2286629 (2008a:47082)
  • 2. H. H. Bauschke, ``Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings'', Proceedings of the American Mathematical Society, vol. 135, pp. 135-139, 2007. MR 2280182 (2008a:49017)
  • 3. H. H. Bauschke, J. M. Borwein, and X. Wang, ``Fitzpatrick functions and continuous linear monotone operators'', SIAM Journal on Optimization, vol. 18, pp. 789-809, 2007. MR 2345969
  • 4. H. H. Bauschke, R. Goebel, Y. Lucet, and X. Wang, ``The proximal average: basic theory'', SIAM Journal on Optimization, vol. 19, pp. 766-785, 2008. MR 2425040
  • 5. H. H. Bauschke, Y. Lucet, and X. Wang, ``Primal-dual symmetric antiderivatives for cyclically monotone operators'', SIAM Journal on Control and Optimization, vol. 46, pp. 2031-2051, 2007.
  • 6. H. H. Bauschke, E. Matoušková, and S. Reich, ``Projection and proximal point methods: convergence results and counterexamples'', Nonlinear Analysis, vol. 56, pp. 715-738, 2004. MR 2036787 (2004m:47116)
  • 7. H. H. Bauschke, D. A. McLaren, and H. S. Sendov, ``Fitzpatrick functions: inequalities, examples and remarks on a problem by S. Fitzpatrick'', Journal of Convex Analysis, vol. 13, pp. 499-523, 2006. MR 2291550 (2007k:49034)
  • 8. J. M. Borwein, ``Maximal monotonicity via convex analysis'', Journal of Convex Analysis, vol. 13, pp. 561-586, 2006. MR 2291552
  • 9. R. S. Burachik and S. Fitzpatrick, ``On the Fitzpatrick family associated to some subdifferentials'', Journal of Nonlinear and Convex Analysis, vol. 6, pp. 165-171, 2005. MR 2138108 (2006b:47081)
  • 10. S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, Australian National University vol. 20, Canberra, Australia, pp. 59-65, 1988. MR 1009594 (90i:47054)
  • 11. N. Ghoussoub, ``Maximal monotone operators are selfdual vector fields and vice-versa'', Proceedings of the American Mathematical Society, to appear.
  • 12. J.-E. Martínez-Legaz and B. F. Svaiter, ``Monotone operators representable by l.s.c. convex functions,'' Set-Valued Analysis, vol. 13, pp. 21-46, 2005. MR 2128696 (2005m:47104)
  • 13. J.-E. Martínez-Legaz and M. Théra, ``A convex representation of maximal monotone operators,'' Journal of Nonlinear and Convex Analysis, vol. 2, pp. 243-247, 2001. MR 1848704 (2002e:49035)
  • 14. J.-P. Penot, ``Autoconjugate functions and representations of monotone operators'', Bulletin of the Australian Mathematical Society, vol. 67, pp. 277-284, 2003. MR 1972717 (2004b:49075)
  • 15. J.-P. Penot, ``The relevance of convex analysis for the study of monotonicity'', Nonlinear Analysis, vol. 58, pp. 855-871, 2004. MR 2086060 (2005g:49026)
  • 16. J.-P. Penot and C. Zălinescu, ``Some problems about the representation of monotone operators by convex functions'', The Australian & New Zealand Industrial and Applied Mathematics Journal, vol. 47, pp. 1-20, 2005. MR 2159848 (2006d:47090)
  • 17. S. Reich and S. Simons, ``Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem'', Proceedings of the American Mathematical Society, vol. 133, pp. 2657-2660, 2005. MR 2146211 (2006d:46025)
  • 18. R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. MR 0274683 (43:445)
  • 19. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, 1998. MR 1491362 (98m:49001)
  • 20. S. Simons, Minimax and Monotonicity, Springer-Verlag, 1998. MR 1723737 (2001h:49002)
  • 21. S. Simons and C. Zălinescu, ``A new proof for Rockafellar's characterization of maximal monotone operators'', Proceedings of the American Mathematical Society, vol. 132, pp. 2969-2972, 2004. MR 2063117 (2005f:47121)
  • 22. S. Simons and C. Zălinescu, ``Fenchel duality, Fitzpatrick functions and maximal monotonicity'', Journal of Nonlinear and Convex Analysis, vol. 6, pp. 1-22, 2005. MR 2138099 (2005k:49102)
  • 23. B. F. Svaiter, ``Fixed points in the family of convex representations of a maximal monotone operator'', Proceedings of the American Mathematical Society, vol. 131, pp. 3851-3859, 2003. MR 1999934 (2004h:49016)
  • 24. C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002. MR 1921556 (2003k:49003)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52A41, 47N10, 47H05

Retrieve articles in all journals with MSC (2000): 52A41, 47N10, 47H05


Additional Information

Heinz H. Bauschke
Affiliation: Department of Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia, Canada V1V 1V7
Email: heinz.bauschke@ubc.ca

Xianfu Wang
Affiliation: Department of Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia, Canada V1V 1V7
Email: shawn.wang@ubc.ca

DOI: https://doi.org/10.1090/S0002-9947-09-04698-4
Keywords: Arithmetic average, autoconjugate, convex function, epigraphical average, Fenchel conjugate, Fitzpatrick function, maximal monotone operator, monotone operator, proximal average, proximal mapping, selfdual Lagrangian
Received by editor(s): May 10, 2007
Received by editor(s) in revised form: October 2, 2007
Published electronically: April 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society