# 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 $u:\left[a,b\right]\to ℝ$ are not necessarily convex and satisfy $u\le f$ on $\left[a,b\right]$, $u\left(a\right)=f\left(a\right)$, $u\left(b\right)=f\left(b\right)$, $f$ is a fixed function on $\left[a,b\right]$. 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 - Recherche Opérationnelle 36.1 (2002): 95-100. <http://eudml.org/doc/245220>.

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 $u:[a,b]\rightarrow \mathbb \{R\}$ 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 $\bar\{f\}$, the convex envelope of $f$.},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {convex envelope; optimization; strict convexity; cost function; minima of functionals},
language = {eng},
number = {1},
pages = {95-100},
publisher = {EDP-Sciences},
title = {Generalized characterization of the convex envelope of a function},
url = {http://eudml.org/doc/245220},
volume = {36},
year = {2002},
}

TY - JOUR
TI - Generalized characterization of the convex envelope of a function
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2002
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 $u:[a,b]\rightarrow \mathbb {R}$ 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 $\bar{f}$, the convex envelope of $f$.
LA - eng
KW - convex envelope; optimization; strict convexity; cost function; minima of functionals
UR - http://eudml.org/doc/245220
ER -

## References

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1. [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. Zbl0876.49018MR1416513
2. [2] H. Brezis, Analyse Fonctionnelle: Théorie et Applications. Masson, Paris, France (1983). Zbl0511.46001MR697382
3. [3] B. Dacorogna, Introduction au Calcul des Variations. Presses Polytechniques et Universitaires Romandes, Lausanne (1992). Zbl0757.49001MR1169677
4. [4] F. Kadhi and A. Trad, Characterization and Approximation of the Convex Envelope of a Function. J. Optim. Theory Appl. 110 (2001) 457-466. Zbl1007.90049MR1846278
5. [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. Zbl0889.47035
6. [6] T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970). Zbl0193.18401MR274683
7. [7] W. Rudin, Real and Complex Analysis, Third Edition. McGraw Hill, New York (1987). Zbl0925.00005MR924157

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