# Generalized Characterization of the Convex Envelope of a Function

RAIRO - Operations Research (2010)

- Volume: 36, Issue: 1, page 95-100
- ISSN: 0399-0559

## Access Full Article

top## Abstract

top## How to cite

topKadhi, Fethi. "Generalized Characterization of the Convex Envelope of a Function." RAIRO - Operations Research 36.1 (2010): 95-100. <http://eudml.org/doc/105263>.

@article{Kadhi2010,

abstract = {
We investigate the minima of functionals of the form $$\int\_\{[a,b]\}g(\dot u(s))\{\rm d\}s$$
where g is strictly convex. The admissible functions $u:[a,b]\longrightarrow\mathbb\{R\}$ are not necessarily
convex and satisfy $u\leq f$ on [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.
},

author = {Kadhi, Fethi},

journal = {RAIRO - Operations Research},

keywords = {Convex envelope; optimization; strict convexity; cost function.; cost function; minima of functionals; convex envelope},

language = {eng},

month = {3},

number = {1},

pages = {95-100},

publisher = {EDP Sciences},

title = {Generalized Characterization of the Convex Envelope of a Function},

url = {http://eudml.org/doc/105263},

volume = {36},

year = {2010},

}

TY - JOUR

AU - Kadhi, Fethi

TI - Generalized Characterization of the Convex Envelope of a Function

JO - RAIRO - Operations Research

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 1

SP - 95

EP - 100

AB -
We investigate the minima of functionals of the form $$\int_{[a,b]}g(\dot u(s)){\rm d}s$$
where g is strictly convex. The admissible functions $u:[a,b]\longrightarrow\mathbb{R}$ are not necessarily
convex and satisfy $u\leq f$ on [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.

LA - eng

KW - Convex envelope; optimization; strict convexity; cost function.; cost function; minima of functionals; convex envelope

UR - http://eudml.org/doc/105263

ER -

## References

top- 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.
- H. Brezis, Analyse Fonctionnelle: Théorie et Applications. Masson, Paris, France (1983). Zbl0511.46001
- B. Dacorogna, Introduction au Calcul des Variations. Presses Polytechniques et Universitaires Romandes, Lausanne (1992).
- F. Kadhi and A. Trad, Characterization and Approximation of the Convex Envelope of a Function. J. Optim. Theory Appl.110 (2001) 457-466. Zbl1007.90049
- 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.
- T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970). Zbl0193.18401
- W. Rudin, Real and Complex Analysis, Third Edition. McGraw Hill, New York (1987). Zbl0925.00005

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.