Generalized characterization of the convex envelope of a function

Kadhi, Fethi

doi:10.1051/ro:2002007

Kadhi, Fethi

RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 1, pp. 95-100.

Résumé

We investigate the minima of functionals of the form

\int_{[a, b]} g (\dot{u} (s)) d s

where

g

is strictly convex. The admissible functions

u : [a, b] \to ℝ

are not necessarily convex and satisfy

u \leq f

[a, b]

u (a) = f (a)

u (b) = f (b)

f

is a fixed function on

[a, b]

. We show that the minimum is attained by

\bar{f}

, the convex envelope of

f

MR Zbl

DOI : 10.1051/ro:2002007

Mots-clés : convex envelope, optimization, strict convexity, cost function

@article{RO_2002__36_1_95_0,
     author = {Kadhi, Fethi},
     title = {Generalized characterization of the convex envelope of a function},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {95--100},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {1},
     year = {2002},
     doi = {10.1051/ro:2002007},
     mrnumber = {1920381},
     zbl = {1003.49016},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro:2002007/}
}

TY  - JOUR
AU  - Kadhi, Fethi
TI  - Generalized characterization of the convex envelope of a function
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2002
SP  - 95
EP  - 100
VL  - 36
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro:2002007/
DO  - 10.1051/ro:2002007
LA  - en
ID  - RO_2002__36_1_95_0
ER  -

%0 Journal Article
%A Kadhi, Fethi
%T Generalized characterization of the convex envelope of a function
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2002
%P 95-100
%V 36
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro:2002007/
%R 10.1051/ro:2002007
%G en
%F RO_2002__36_1_95_0

Kadhi, Fethi. Generalized characterization of the convex envelope of a function. RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 1, pp. 95-100. doi : 10.1051/ro:2002007. http://www.numdam.org/articles/10.1051/ro:2002007/

Bibliographie
Cité par

[1] J. Benoist and J.B. Hiriart-Urruty, What Is the Subdifferential of the Closed Convex Hull of a Function? SIAM J. Math. Anal. 27 (1994) 1661-1679. | MR | Zbl

[2] H. Brezis, Analyse Fonctionnelle: Théorie et Applications. Masson, Paris, France (1983). | MR | Zbl

[3] B. Dacorogna, Introduction au Calcul des Variations. Presses Polytechniques et Universitaires Romandes, Lausanne (1992). | MR | Zbl

[4] F. Kadhi and A. Trad, Characterization and Approximation of the Convex Envelope of a Function. J. Optim. Theory Appl. 110 (2001) 457-466. | MR | Zbl

[5] T. Lachand-Robert and M.A. Peletier, Minimisation de Fonctionnelles dans un Ensemble de Fonctions Convexes. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 851-855. | Zbl

[6] T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970). | MR | Zbl

[7] W. Rudin, Real and Complex Analysis, Third Edition. McGraw Hill, New York (1987). | MR | Zbl

Cité par Sources :