Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-27T00:31:57.686Z Has data issue: false hasContentIssue false
  • English
  • Français

When some variational properties force convexity

Published online by Cambridge University Press:  17 May 2013

M. Volle ,
J.-B. Hiriart-Urruty  and
C. Zălinescu
M. Volle
Affiliation:
Department of Mathematics, University of Avignon, France. michel.volle@univ-avignon.fr
J.-B. Hiriart-Urruty
Affiliation:
Institut de Mathématiques, Université Paul Sabatier, Toulouse, France; jbhu@cict.fr
C. Zălinescu
Affiliation:
Faculty of Mathematics, University Al. I. Cuza, Iaşi, Romania Octav Mayer Institute of Mathematics, Romanian Academy, Romania; zalinesc@uaic.ro
Get access

Abstract

The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach.

Keywords

Convex dualitywell posed optimization problemessential strict convexityessential smoothnessbest approximation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 701 - 709
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asplund, E., Fréchet differentiability of convex functions. Acta Math. 121 (1968) 31–47. Google Scholar
Asplund, E., Čebysev sets in Hilbert spaces. Trans. Amer. Math. Soc. 9 (1969) 235–240. Google Scholar
Asplund, E., Differentiability of the metric projection in finite-dimensional Euclidean spaces. Proc. Amer. Math. Soc. 38 (1973) 218–219. Google Scholar
Asplund, E. and Rockafellar, R.T., Gradients of convex functions. Trans. Amer. Math. Soc. 139 (1969) 443–467. Google Scholar
Bauschke, H.H., Borwein, J.M. and Combettes, P.L., Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3 (2001) 615–647. Google Scholar
Blatter, J., Weiteste Punkte und nächste Punkte. Rev. Roum. Math. Pures Appl. 14 (1969) 615–621. Google Scholar
J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and its Applications, vol. 109. Cambridge University Press, Cambridge (2010).
Clarke, F.H., Stern, R.J. and Wolenski, P.R., Proximal smoothness and the lower-C2 property. J. Conv. Anal. 2 (1995) 117–144. Google Scholar
Cobzaş, S., Geometric properties of Banach spaces and the existence of nearest and farthest points. Abstract Appl. Anal. 3 (2005) 259–285. Google Scholar
A.L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems. Springer-Verlag, Berlin (1993).
Efimov, N.V. and Steckin, S.B., Approximative compactness and Čebysev sets. Soviet Math. Dokl. 2 (1961) 1226–1228. Google Scholar
Fitzpatrick, S., Metric projections and the differentiability of distance functions. Bull. Austral. Math. Soc. 22 (1980) 291–312. Google Scholar
Hiriart-Urruty, J.-B., Ensembles de Tchebychev vs. ensembles convexes: l’état de la situation vu via l’analyse convexe non lisse. Ann. Sc. Math. Québec 22 (1998) 47–62. Google Scholar
Hiriart-Urruty, J.-B., La conjecture des points les plus eloignés revisitée. Ann. Sci. Math. Québec 29 (2005) 197–214. Google Scholar
Klee, V., Convexity of Chebyshev sets. Math. Ann. 142 (1961) 292–304 Google Scholar
Lassonde, M., Asplund spaces, Stegall variational principle and the RNP. Set-Valued Var. Anal. 17 (2009) 183–193. Google Scholar
J.-J. Moreau, Fonctionnelles Convexes, Collège de France, 1966. Republished by the “Tor Vergata” University, Rome (2003).
T.D., Narang, A study of farthest points. Nieuw Arch. Voor Wiscunde 3 (1977) XXV 54–79. Google Scholar
Panda, B.B. and Kapoor, O.P., On farthest points of sets. J. Math. Anal. Appl. 62 (1978) 345–353. Google Scholar
R.R. Phelps, Convex Functions, Monotone Operators and Differentiability. Lect. Notes Math., vol. 1364. Springer-Verlag (1989).
T., Precupanu, Relationships between farthest point problem and best approximation problem. Anal. Sci. Univ. AI. I. Cuza, Mat. 57 (2011) 1–12. Google Scholar
Soloviov, V., Duality for nonconvex optimization and its applications. Anal. Math. 19 (1993) 297–315. Google Scholar
V. Soloviov, Characterization of convexity in terms of smoothness. Unpublished report, Moscow Aviation Institute (1995).
Stromberg, T., Duality between Fréchet differentiability and strong convexity. Positivity 15 (2011) 527–536. Google Scholar
Volle, M. and Hiriart-Urruty, J.-B., A characterization of essentially strictly convex functions in reflexive Banach spaces. Nonlinear Anal. 75 (2012) 1617–1622. Google Scholar
M. Volle and C. Zălinescu, On strongly adequate functions on Banach spaces. J. Convex Anal. (to appear).
Wang, X., On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368 (2010) 293–310. Google Scholar
C. Zălinescu, Convex Analysis in General Vector Spaces. World Scientific, River Edge, N.J. (2002).