# Generalized Characterization of the Convex Envelope of a Function

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

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## Abstract

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We investigate the minima of functionals of the form ${\int }_{\left[a,b\right]}g\left(\stackrel{˙}{u}\left(s\right)\right)\mathrm{d}s$ where g is strictly convex. The admissible functions $u:\left[a,b\right]⟶ℝ$ are not necessarily convex and satisfy $u\le f$ 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 $\overline{f}$, the convex envelope of f.

## How to cite

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

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. },
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
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

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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.
2. H. Brezis, Analyse Fonctionnelle: Théorie et Applications. Masson, Paris, France (1983).  Zbl0511.46001
3. B. Dacorogna, Introduction au Calcul des Variations. Presses Polytechniques et Universitaires Romandes, Lausanne (1992).
4. 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
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.
6. T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970).  Zbl0193.18401
7. W. Rudin, Real and Complex Analysis, Third Edition. McGraw Hill, New York (1987).  Zbl0925.00005

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