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A numerical approach to variational problems subject to convexity constraint

Carlier, Guillaume; Lachand-Robert, Thomas; Maury, Bertrand (2001), A numerical approach to variational problems subject to convexity constraint, Numerische Mathematik, 88, 2, p. 299-318. http://dx.doi.org/10.1007/PL00005446

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Type
Article accepté pour publication ou publié
Date
2001
Journal name
Numerische Mathematik
Volume
88
Number
2
Publisher
Springer
Pages
299-318
Publication identifier
http://dx.doi.org/10.1007/PL00005446
Metadata
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Author(s)
Carlier, Guillaume
Lachand-Robert, Thomas
Maury, Bertrand
Abstract (EN)
We describe an algorithm to approximate the minimizer of an elliptic functional in the form R Ω j(x, u,∇u) on the set C of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope u∗∗ 0 of a given function u0. Let (Tn) be any quasiuniform sequence of meshes whose diameter goes to zero, and In the corresponding affine interpolation operators. We prove that the minimizer over C is the limit of the sequence (un), where un minimizes the functional over In(C). We give an implementable characterization of In(C). Then the finite dimensional problem turns out to be a minimization problem with linear constraints.
Subjects / Keywords
Linear constraints; Minimization problems; convexity constraint

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