Geodesic
A Necessary and Sufficient Optimality Condition for a Class of Nonconvex Scalar Variational Problems
Journal of convex analysis, Tome 11 (2004) no. 2, pp. 401-411.

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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.
Mots-clés : convexity constraint, monotone rearrangements, duality 
@article{JCA_2004_11_2_JCA_2004_11_2_a8,
     author = {G. Carlier},
     title = {A {Necessary} and {Sufficient} {Optimality} {Condition} for a {Class} of {Nonconvex} {Scalar} {Variational} {Problems}},
     journal = {Journal of convex analysis},
     pages = {401--411},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2004},
     url = {http://geodesic.mathdoc.fr/item/JCA_2004_11_2_JCA_2004_11_2_a8/}
}
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G. Carlier. A Necessary and Sufficient Optimality Condition for a Class of Nonconvex Scalar Variational Problems. Journal of convex analysis, Tome 11 (2004) no. 2, pp. 401-411. http://geodesic.mathdoc.fr/item/JCA_2004_11_2_JCA_2004_11_2_a8/