Generalized characterization of the convex envelope of a function

Kadhi, Fethi

doi:10.1051/ro:2002007

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 (\dot{u} (s)) d s

where

g

is strictly convex. The admissible functions

u : [a, b] \to 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

.

MR Zbl

DOI : 10.1051/ro:2002007

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

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     author = {Kadhi, Fethi},
     title = {Generalized characterization of the convex envelope of a function},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {95--100},
     year = {2002},
     publisher = {EDP Sciences},
     volume = {36},
     number = {1},
     doi = {10.1051/ro:2002007},
     mrnumber = {1920381},
     zbl = {1003.49016},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ro:2002007/}
}

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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

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