Generalized characterization of the convex envelope of a function
RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 1, pp. 95-100.

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 $f$.

DOI : https://doi.org/10.1051/ro:2002007
Mots clés : convex envelope, optimization, strict convexity, cost function
@article{RO_2002__36_1_95_0,
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},
zbl = {1003.49016},
mrnumber = {1920381},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2002007/}
}
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/

[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 1416513 | Zbl 0876.49018

[2] H. Brezis, Analyse Fonctionnelle: Théorie et Applications. Masson, Paris, France (1983). | MR 697382 | Zbl 0511.46001

[3] B. Dacorogna, Introduction au Calcul des Variations. Presses Polytechniques et Universitaires Romandes, Lausanne (1992). | MR 1169677 | Zbl 0757.49001

[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 1846278 | Zbl 1007.90049

[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. | Zbl 0889.47035

[6] T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, New Jersey (1970). | MR 274683 | Zbl 0193.18401

[7] W. Rudin, Real and Complex Analysis, Third Edition. McGraw Hill, New York (1987). | MR 924157 | Zbl 0925.00005