Generalized characterization of the convex envelope of a function

Fethi Kadhi

RAIRO - Operations Research - Recherche Opérationnelle (2002)

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

Abstract

top
We investigate the minima of functionals of the form ∫ [ a , b ] g ( u ˙ ( s ) ) d s where g is strictly convex. The admissible functions u : [ a , b ] → ℝ are not necessarily convex and satisfy u ≤ 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 f ¯ , the convex envelope of f .

How to cite

top

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>.

@article{Kadhi2002,
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 $f$.},
author = {Kadhi, Fethi},
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
AU - Kadhi, Fethi
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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.