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 duality, well posed optimization problem, essential strict convexity, essential smoothness, best approximation
@article{COCV_2013__19_3_701_0, author = {Volle, M. and Hiriart-Urruty, J.-B. and Z\u{a}linescu, C.}, title = {When some variational properties force convexity}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {701--709}, publisher = {EDP-Sciences}, volume = {19}, number = {3}, year = {2013}, doi = {10.1051/cocv/2012029}, mrnumber = {3092358}, zbl = {1277.46042}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012029/} }
TY - JOUR AU - Volle, M. AU - Hiriart-Urruty, J.-B. AU - Zălinescu, C. TI - When some variational properties force convexity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 701 EP - 709 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012029/ DO - 10.1051/cocv/2012029 LA - en ID - COCV_2013__19_3_701_0 ER -
%0 Journal Article %A Volle, M. %A Hiriart-Urruty, J.-B. %A Zălinescu, C. %T When some variational properties force convexity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 701-709 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012029/ %R 10.1051/cocv/2012029 %G en %F COCV_2013__19_3_701_0
Volle, M.; Hiriart-Urruty, J.-B.; Zălinescu, C. When some variational properties force convexity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 701-709. doi : 10.1051/cocv/2012029. http://www.numdam.org/articles/10.1051/cocv/2012029/
[1] Fréchet differentiability of convex functions. Acta Math. 121 (1968) 31-47. | MR | Zbl
,[2] Čebysev sets in Hilbert spaces. Trans. Amer. Math. Soc. 9 (1969) 235-240. | MR | Zbl
,[3] Differentiability of the metric projection in finite-dimensional Euclidean spaces. Proc. Amer. Math. Soc. 38 (1973) 218-219. | MR | Zbl
,[4] Gradients of convex functions. Trans. Amer. Math. Soc. 139 (1969) 443-467. | MR | Zbl
and ,[5] Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3 (2001) 615-647. | MR | Zbl
, and ,[6] Weiteste Punkte und nächste Punkte. Rev. Roum. Math. Pures Appl. 14 (1969) 615-621. | MR | Zbl
,[7] Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and its Applications, vol. 109. Cambridge University Press, Cambridge (2010). | MR | Zbl
and ,[8] Proximal smoothness and the lower-C2 property. J. Conv. Anal. 2 (1995) 117-144. | MR | Zbl
, and ,[9] Geometric properties of Banach spaces and the existence of nearest and farthest points. Abstract Appl. Anal. 3 (2005) 259-285. | MR | Zbl
,[10] Well-Posed Optimization Problems. Springer-Verlag, Berlin (1993). | MR | Zbl
and ,[11] Approximative compactness and Čebysev sets. Soviet Math. Dokl. 2 (1961) 1226-1228. | Zbl
and ,[12] Metric projections and the differentiability of distance functions. Bull. Austral. Math. Soc. 22 (1980) 291-312. | MR | Zbl
,[13] 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. | MR | Zbl
,[14] La conjecture des points les plus eloignés revisitée. Ann. Sci. Math. Québec 29 (2005) 197-214. | MR | Zbl
,[15] Convexity of Chebyshev sets. Math. Ann. 142 (1961) 292-304 | MR | Zbl
,[16] Asplund spaces, Stegall variational principle and the RNP. Set-Valued Var. Anal. 17 (2009) 183-193. | MR | Zbl
,[17] Fonctionnelles Convexes, Collège de France, 1966. Republished by the “Tor Vergata” University, Rome (2003).
,[18] A study of farthest points. Nieuw Arch. Voor Wiscunde 3 (1977) XXV 54-79. | MR | Zbl
,[19] On farthest points of sets. J. Math. Anal. Appl. 62 (1978) 345-353. | MR | Zbl
and ,[20] Convex Functions, Monotone Operators and Differentiability. Lect. Notes Math., vol. 1364. Springer-Verlag (1989). | MR | Zbl
,[21] Relationships between farthest point problem and best approximation problem. Anal. Sci. Univ. AI. I. Cuza, Mat. 57 (2011) 1-12. | MR | Zbl
,[22] Duality for nonconvex optimization and its applications. Anal. Math. 19 (1993) 297-315. | MR | Zbl
,[23] Characterization of convexity in terms of smoothness. Unpublished report, Moscow Aviation Institute (1995).
,[24] Duality between Fréchet differentiability and strong convexity. Positivity 15 (2011) 527-536. | MR | Zbl
,[25] A characterization of essentially strictly convex functions in reflexive Banach spaces. Nonlinear Anal. 75 (2012) 1617-1622. | MR | Zbl
and ,[26] On strongly adequate functions on Banach spaces. J. Convex Anal. (to appear). | MR | Zbl
and ,[27] On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368 (2010) 293-310. | MR | Zbl
,[28] Convex Analysis in General Vector Spaces. World Scientific, River Edge, N.J. (2002). | Zbl
,Cited by Sources: