A Necessary and Sufficient Optimality Condition for a Class of Nonconvex Scalar Variational Problems

Guillaume Carlier

doi:10.68381/jca11025

Abstract

This article studies the minimization of the functional

u\mapsto\int_{0}^{1}f(\dot{u})

among all convex functions

u

that satisfy the additional obstacle constraint

u\geq \ovu

,

u(0)=\ovu(0)

,

u(1)=\ovu(1)

where

\ovu

is a given convex function. We first show that this nonconvex problem is in fact equivalent to a linear programming problem. This enables us to establish a necessary and sufficient optimality condition.

Author information

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

Guillaume Carlier

Université Bordeaux IV, GRAPE, UMR CNRS 5113, Avenue Léon Duguit, 33608 Pessac, France
and: Université Bordeaux 1, MAB, UMR CNRS 5466, France

Guillaume.Carlier@math.u-bordeaux.fr

Keywords

convexity constraint
monotone rearrangements
duality

References

Y. Brenier: Polar factorization and monotone rearrangements of vector valued functions, Commun. Pure Appl. Math. 44 (1991) 375–417.https://doi.org/10.1002/cpa.3160440402 Google Scholar
F. Brock, V. Ferone, B. Kawohl: A symmetry problem in the calculus of variations, Calc. Var. Partial Differ. Equ. 4 (1996) 71–89.https://doi.org/10.1007/BF01261764 Google Scholar
I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland (1972).Google Scholar
T. Lachand-Robert, M. A. Peletier: Extremal points of a functional on the set of convex functions, Proc. Amer. Math. Soc. 127 (1999) 1723–1727.https://doi.org/10.1090/S0002-9939-99-05209-0 Google Scholar
S. Mazur: Über konvexe Mengen in linearen normierten Räumen, Stud. Math. 4 (1933) 70–84.https://doi.org/10.4064/sm-4-1-70-84 Google Scholar
P.-A. Meyer: Probabilités et Potentiel, Hermann, Paris (1966).Google Scholar
J. Mossino: Inégalités Isopérimétriques et Applications en Physique, Hermann, Paris (1984).Google Scholar
R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lect. Notes Math. 1364, 2nd Ed., Springer (1993).Google Scholar
J.-C. Rochet, P. Choné: Ironing, sweeping and multidimensional screening, Econometrica 66 (1998) 783–826.https://doi.org/10.2307/2999574 Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton University Press (1970).https://doi.org/10.1515/9781400873173 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.

[1] Y. Brenier: Polar factorization and monotone rearrangements of vector valued functions, Commun. Pure Appl. Math. 44 (1991) 375–417.https://doi.org/10.1002/cpa.3160440402 Google Scholar

[2] F. Brock, V. Ferone, B. Kawohl: A symmetry problem in the calculus of variations, Calc. Var. Partial Differ. Equ. 4 (1996) 71–89.https://doi.org/10.1007/BF01261764 Google Scholar

[3] I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland (1972).Google Scholar

[4] T. Lachand-Robert, M. A. Peletier: Extremal points of a functional on the set of convex functions, Proc. Amer. Math. Soc. 127 (1999) 1723–1727.https://doi.org/10.1090/S0002-9939-99-05209-0 Google Scholar

[5] S. Mazur: Über konvexe Mengen in linearen normierten Räumen, Stud. Math. 4 (1933) 70–84.https://doi.org/10.4064/sm-4-1-70-84 Google Scholar

[6] P.-A. Meyer: Probabilités et Potentiel, Hermann, Paris (1966).Google Scholar

[7] J. Mossino: Inégalités Isopérimétriques et Applications en Physique, Hermann, Paris (1984).Google Scholar

[8] R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lect. Notes Math. 1364, 2nd Ed., Springer (1993).Google Scholar

[9] J.-C. Rochet, P. Choné: Ironing, sweeping and multidimensional screening, Econometrica 66 (1998) 783–826.https://doi.org/10.2307/2999574 Google Scholar

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