Generalized characterization of the convex envelope of a function

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

$${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$

where $g$ is strictly convex. The admissible functions $u:[a,b]\to ℝ$ are not necessarily convex and satisfy $u\le f$ on $[a,b]$, $u\left(a\right)=f\left(a\right)$, $u\left(b\right)=f\left(b\right)$, $f$ is a fixed function on $[a,b]$. We show that the minimum is attained by $\overline{f}$, the convex envelope of $f$.

Détail

DOI : 10.1051/ro:2002007

Keywords: convex envelope, optimization, strict convexity, cost function