A numerical approach to variational problems subject to convexity constraint

Carlier, Guillaume; Lachand-Robert, Thomas; Maury, Bertrand

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

Number

Publisher

Springer

Pages

299-318

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