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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Integration of multivalued operators and cyclic submonotonicity


Authors: Aris Daniilidis, Pando Georgiev and Jean-Paul Penot
Journal: Trans. Amer. Math. Soc. 355 (2003), 177-195
MSC (2000): Primary 49J52, 47H05; Secondary 58C20
Published electronically: September 6, 2002
MathSciNet review: 1928084
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Abstract: We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space $X$ to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset $U$ of $X$, then it is also maximal cyclically submonotone on $U$, and, conversely, that every maximal cyclically submonotone operator on $U$ is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if $U$ is connected. In finite dimensions these functions are exactly the lower C$^{1}$ functions considered by Spingarn and Rockafellar.


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  • 1. BIRGE, R. AND QI, L., Semi-regularity and generalized subdifferentials with applications to optimization, Math. Oper. Res. 18 (1993), 982-1005. MR 94h:49026
  • 2. BORWEIN, J. M., Minimal cuscos and subgradients of Lipschitz functions, in: Fixed Point Theory and its Applications, (J.-B. Baillon and M. Théra, eds.), Pitman Res. Notes in Math. Series, No. 252, Longman, Essex, (1991), 57-82. MR 92j:46077
  • 3. BORWEIN, J. AND MOORS, W., Essentially smooth Lipschitz functions, J. Funct. Anal. 149 (1997), 305-351. MR 98i:58028
  • 4. BORWEIN, J., MOORS, W. AND SHAO, Y., Subgradient representation of multifunctions, J. Austral. Math. Soc. (Series B) 40 (1998), 301-313. MR 2001b:49020
  • 5. BORWEIN, J. AND ZHU, Q., Multivalued and functional analytic techniques in nonsmooth analysis, (F. H. Clarke and R. J. Stern, eds.), Nonlinear Analysis, Differential Equations and Control (1999), 61-157. MR 2002a:49016
  • 6. CLARKE, F. H., Optimization and Nonsmooth Analysis, Wiley Interscience, New York (1983). MR 85m:49002
  • 7. CORREA, R. AND JOFRE, A., Tangentially continuous directional derivatives in nonsmooth analysis, J. Opt. Th. Appl. 61 (1989), 1-21. MR 90h:49009
  • 8. CORREA, R. AND THIBAULT, L., Subdifferential analysis of bivariate separately regular functions, J. Math. Anal. Appl. 148 (1990), 157-174. MR 91b:49018
  • 9. DANIILIDIS, A. AND HADJISAVVAS, N., On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity, J. Math. Anal. Appl. 237 (1999), 30-42. MR 2000h:49026
  • 10. GEORGIEV, P., Submonotone mappings in Banach spaces and differentiability of nonconvex functions, Compt. Rend. Acad. Bulg. Sci. 42 (1989), 13-16. MR 90k:58013
  • 11. GEORGIEV, P., Submonotone mappings in Banach spaces and applications, Set-Valued Analysis 5 (1997), 1-35. MR 98d:49021
  • 12. JANIN, R., Sur des multiapplications qui sont des gradients généralisés, C.R. Acad. Sci. Paris 294 (1982), 117-119. MR 83d:58013
  • 13. LEBOURG, G., Generic differentiability of Lipschitzian functions, Trans. Amer. Math. Soc. 256 (1979), 125-144. MR 80i:58012
  • 14. MIFFLIN, R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim. 15 (1977), 959-972. MR 57:1541
  • 15. MOORS, W., A characterization of minimal subdifferential mappings of locally Lipschitz functions, Set-Valued Analysis 3 (1995), 129-141. MR 96e:58013
  • 16. PENOT, J.-P., Favorable classes of mappings and multimappings in nonlinear analysis and optimization, J. Convex Analysis 3 (1996), 97-116. MR 97i:90110
  • 17. POLIQUIN, R., Integration of subdifferentials of nonconvex functions, Nonlinear Analysis TMA 17 (1991), 385-398. MR 92i:49008
  • 18. PREISS, D., Differentiability of Lipschitz functions on Banach spaces, J. Functional Analysis 91 (1990), 312-345. MR 91g:46051
  • 19. QI, L., The maximal normal operator space and integration of subdifferentials of nonconvex functions, Nonlinear Analysis TMA 13 (1989), 1003-1011. MR 91a:90150
  • 20. ROCKAFELLAR, R. T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216. MR 41:7432
  • 21. ROCKAFELLAR, R. T., ``Favorable classes of Lipschitz continuous functions in subgradient optimization'' in Nondifferentiable Optimization (1982), Nurminski E. (ed.), Pergamon Press, New York. MR 85e:90069
  • 22. ROCKAFELLAR, R. T. AND WETS, J.-B., Variational Analysis, Springer, New York (1998). MR 98m:49001
  • 23. PREISS, D., PHELPS, R. AND NAMIOKA, I., Smooth Banach spaces, weak Asplund spaces and monotone or USCO mappings, Israel J. Math. 72 (1990), 257-279. MR 92h:46021
  • 24. SPINGARN, J. E., Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc. 264 (1981), 77-89. MR 82g:26016
  • 25. THIBAULT, L. AND ZAGRODNY, D., Integration of subdifferentials of lower semi-continuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), 33-58. MR 95i:49032
  • 26. WANG, X., Fine and pathological properties of subdifferentials, Ph.D. Dissertation (1999), Simon Fraser University, Vancouver, Canada.
  • 27. WU, Z. AND YE, J., Some results on integration of subdifferentials, Nonlinear Analysis TMA 39 (2000), 955-976. MR 2000k:49022

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Additional Information

Aris Daniilidis
Affiliation: Laboratoire de Mathématiques Appliquées, CNRS ERS 2055, Université de Pau et des Pays de l’Adour, avenue de l’Université, 64000 Pau, France
Address at time of publication: CODE - Edifici B, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain
Email: aris.daniilidis@univ-pau.fr

Pando Georgiev
Affiliation: Sofia University “St. Kl. Ohridski”, Faculty of Mathematics and Informatics, 5 J. Bourchier Blvd., 1126 Sofia, Bulgaria
Address at time of publication: Laboratory for Advanced Brain Signal Processing, Brain Science Institute, The Institute of Physical and Chemical Research (RIKEN), 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan
Email: georgiev@bsp.brain.riken.go.jp

Jean-Paul Penot
Affiliation: Laboratoire de Mathématiques Appliquées, CNRS ERS 2055, Université de Pau et des Pays de l’Adour, avenue de l’Université, 64000 Pau, France
Email: jean-paul.penot@univ-pau.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03118-5
PII: S 0002-9947(02)03118-5
Keywords: Integration, subdifferential, submonotone operator, subsmooth function
Received by editor(s): May 4, 2000
Published electronically: September 6, 2002
Additional Notes: The research of the first author was supported by the TMR grant ERBFMBI CT 983381
A major part of this work was accomplished while the second author was visiting the University of Pau under the NATO\ grant CB/JB SC105 N$^{0}$ 44/96165
Article copyright: © Copyright 2002 American Mathematical Society