Abstract

We consider optimization problems for a class of convex functions on H × H introduced by Simon Fitzpatrick, where H is a real Hilbert space. We show that the minimization problem of Fitzpatrick functions can be transformed from solving of the correspondent differential inclusions (d.i) on H × H, to solving simplified d.i. on H. By using the idea of optimization of Fitzpatrick functions we introduce a numerical algorithm for solving convex smooth optimization problems by reducing the number of the independent variables. We present a comparative study with numerical examples. Finally, we show that Fitzpatrick functions are closely related to Lyapunov functions.

Author information

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

M. Zuhair Nashed

University of Central Florida, Orlando, U.S.A.

Keywords

  • Maximal monotone operator
  • lower semicontinuous convex maps
  • differential inclusions
  • optimization problems

Mathematics Subject Classification

46N10, 47N10

References

  1. J. P. Aubin, I. Ekeland: Applied Nonlinear Analysis, Wiley-Interscience, New York (1984).Google Scholar
  2. S. Bartz, H. H. Bauschke, J. M. Borwein, S. Reich, X. Wang: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative, Nonlinear Analysis 66 (2007) 1198–1223.https://doi.org/10.1016/j.na.2006.01.013Google Scholar
  3. H. H. Bauschke, D. A. McLaren, H. S. Sendov: Fitzpatrick functions: Inequalities, examples, and remarks on a problem by S. Fitzpatrick, J. Convex Analysis 13/3 (2006) 499–523.https://doi.org/10.68381/jca13043Google Scholar
  4. M. S. Bazara, J. J. Goode, M. Z. Nashed: On the cones of tangents with applications to mathematical programming, 13/4 (1974) 389–426.https://doi.org/10.1007/BF00934938Google Scholar
  5. J. M. Borwein, J. Dutta: Maximal monotone inclusions and Fitzparick functions, J. Optimization Theory Appl. 171/3 (2016) 757–784.https://doi.org/10.1007/s10957-015-0813-xGoogle Scholar
  6. F. H. Clarke: Optimization and Nonsmooth Analysis, Wiley & Sons, New York (1983).Google Scholar
  7. K. Deimling: Multivalued Differential Equations, de Gruyer, Berlin (1992).https://doi.org/10.1515/9783110874228Google Scholar
  8. A. F. Filippov: Differential Equations with Discontinues Right Hand Side (Russian), Nauka, Moscow (1985).Google Scholar
  9. S. Fitzpatrick: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization, Canberra 1988, Proc. Centre for Mathematical Analysis 20, Australian National University, Canberra (1988) 59–65.Google Scholar
  10. R. P. Ivanov, I. Raykov: Parametric Lyapunov function method for solving nonlinear systems in Hilbert spaces, Numer. Funct. Analysis Optimization 17 (1996) 893–901.https://doi.org/10.1080/01630569608816732Google Scholar
  11. N. Pavel, I. Raykov: Parametric proximal-point algorithms, J. Optimization Theory Appl. 139/1 (2008) 85–107.https://doi.org/10.1007/s10957-008-9408-0Google Scholar
  12. J.-P. Penot: The relevance of convex analysis for the study of monotonicity, Nonlinear Analysis Theory Meth. Appl. 58 (2004) 855–871.https://doi.org/10.1016/j.na.2004.05.018Google Scholar
  13. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  14. S. Simons, C. Zălinescu: Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Analysis 6/1 (2005) 1–22.Google Scholar

Suggested citation

I. Raykov, M. Z. Nashed. “Optimization of Fitzpatrick Functions and a Numerical Minimization Algorithm.” Journal of Convex Analysis 28 (2021), No. 4, 1119–1136. https://doi.org/10.68381/jca28065

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.