Normality and Quasiconvex Integrands

Abdessamad Amir; Hocine Mokhtar-Kharroubi

doi:10.68381/jca17005

Abstract

Let

(T, \mathcal{A})

be an arbitrary measurable space and

f

an integrand defined on

T\times \mathbb{R}^n

such that

f(t, \cdot)

is quasiconvex and lower semicontinuous. Here, convexity is present by the level set mapping. We show that the normality property of the integrand in the sense of R. T. Rockafellar [Pacific Journal of Mathematics 24 (1968) 525–539; and in: Nonlinear Operators and the Calculus of Variations; Bruxelles 1975, Lecture Notes in Mathematics 543, 157–207, Springer, Berlin] can be characterized by the normality of the level set mapping, and that normality is preserved for quasiconvex conjugates. Finally we obtain for the integral

I_f (x(\cdot)) = \int_T f(t, x(t)) d\mu (t)

the equality (in appropriate topology) between the lower semicontinuous regularization and the second quasiconvex conjugate.

Author information

Contact details are reproduced from the original publication and may be historical.

Abdessamad Amir

University of Mostaganem, Faculty of Sciences, Dept. of Mathematics, 27000 Mostaganem, Algeria

amir@univ-mosta.dz

Hocine Mokhtar-Kharroubi

University of Oran, Faculty of Sciences, Dept. of Mathematics, Oran, Algeria

hmkharroubi@yahoo.fr

Keywords

Normal integrand
quasiconvex functions
conjugation

Mathematics Subject Classification

26B25,49N15, 49J53

References

E. N. Barron, W. Liu: Calculus of variations in L∞, Appl. Math. Optimization 35 (1997) 213–267.https://doi.org/10.1007/BF02683330 Google Scholar
C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685 Google Scholar
J. P. Crouzeix: Contributions à l’étude des fonctions quasi-convexes, PhD Thesis, Université de Clermont-Ferrand 2 (1977).Google Scholar
J. P. Crouzeix: Continuity and differentiability properties of quasiconvex functions on Rn, in: Generalized Concavity in Optimization and Economics (Vancouver, 1980), S. Schaible, W. T. Ziemba (eds.), Academic Press, New York (1981) 109–130.Google Scholar
I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Etudes Mathématiques, Dunod Gauthier-Villars, Paris (1974).Google Scholar
J. B. G. Frenk, G. Kassay: Introduction to convex and quasiconvex analysis, in: Handbook of Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, S. Komlosi, S. Schaible (eds.), Springer, New York (2005) 3–87.https://doi.org/10.1007/0-387-23393-8_1 Google Scholar
J. E. Martínez-Legaz: Quasiconvex duality theory by generalized conjugation methods, Optimization 19 (1988) 603–652.https://doi.org/10.1080/02331938808843379 Google Scholar
J. J. Moreau: Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl., IX. Sér. 49 (1970) 109–154.Google Scholar
J. P. Penot, M. Volle: On quasiconvex duality, Math. Oper. Res. 15(4) (1990) 597–624.https://doi.org/10.1287/moor.15.4.597 Google Scholar
R. T. Rockafellar: Integrals which are convex functionals, Pac. J. Math. 24(3) (1968) 525–539.https://doi.org/10.2140/pjm.1968.24.525 Google Scholar
R. T. Rockafellar: Integral functionals, normal integrands and measurable selections, in: Nonlinear Operators and the Calculus of Variations (Bruxelles, 1975), J. P. Gossez et al. (ed.), Lecture Notes in Mathematics 543, Springer, Berlin (1976) 157–207.https://doi.org/10.1007/BFb0079944 Google Scholar
R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar
I. Singer: Conjugate operators, in: Selected Topics in Operations Research and Mathematical Economics (Karlsruhe, 1983), G. Hammer, D. Pallaschke (eds.), Springer, Berlin (1984) 80–97.https://doi.org/10.1007/978-3-642-45567-4_6 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.

[1] E. N. Barron, W. Liu: Calculus of variations in L∞, Appl. Math. Optimization 35 (1997) 213–267.https://doi.org/10.1007/BF02683330 Google Scholar

[2] C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685 Google Scholar

[3] J. P. Crouzeix: Contributions à l’étude des fonctions quasi-convexes, PhD Thesis, Université de Clermont-Ferrand 2 (1977).Google Scholar

[4] J. P. Crouzeix: Continuity and differentiability properties of quasiconvex functions on Rn, in: Generalized Concavity in Optimization and Economics (Vancouver, 1980), S. Schaible, W. T. Ziemba (eds.), Academic Press, New York (1981) 109–130.Google Scholar

[5] I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Etudes Mathématiques, Dunod Gauthier-Villars, Paris (1974).Google Scholar

[6] J. B. G. Frenk, G. Kassay: Introduction to convex and quasiconvex analysis, in: Handbook of Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, S. Komlosi, S. Schaible (eds.), Springer, New York (2005) 3–87.https://doi.org/10.1007/0-387-23393-8_1 Google Scholar

[7] J. E. Martínez-Legaz: Quasiconvex duality theory by generalized conjugation methods, Optimization 19 (1988) 603–652.https://doi.org/10.1080/02331938808843379 Google Scholar

[8] J. J. Moreau: Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl., IX. Sér. 49 (1970) 109–154.Google Scholar

[9] J. P. Penot, M. Volle: On quasiconvex duality, Math. Oper. Res. 15(4) (1990) 597–624.https://doi.org/10.1287/moor.15.4.597 Google Scholar

[10] R. T. Rockafellar: Integrals which are convex functionals, Pac. J. Math. 24(3) (1968) 525–539.https://doi.org/10.2140/pjm.1968.24.525 Google Scholar

[11] R. T. Rockafellar: Integral functionals, normal integrands and measurable selections, in: Nonlinear Operators and the Calculus of Variations (Bruxelles, 1975), J. P. Gossez et al. (ed.), Lecture Notes in Mathematics 543, Springer, Berlin (1976) 157–207.https://doi.org/10.1007/BFb0079944 Google Scholar

[12] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar

[13] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar

[14] I. Singer: Conjugate operators, in: Selected Topics in Operations Research and Mathematical Economics (Karlsruhe, 1983), G. Hammer, D. Pallaschke (eds.), Springer, Berlin (1984) 80–97.https://doi.org/10.1007/978-3-642-45567-4_6 Google Scholar