A numerical approach to variational problems subject to convexity constraint

We describe an algorithm to approximate the minimizer of an elliptic functi onal in the form integral(Omega) j(x, u, del u) on the set C of convex func tions u in an appropriate functional space X. Such problems arise for insta nce in mathematical economics [5]. A special case gives the convex envelope u(o)** of a given function u(o). Let (T-n) be arty quasi-uniform sequence of meshes whose diameter goes to zero, and I-n the corresponding affine int erpolation operators. We prove that the minimizer over C is the limit of th e sequence (u(n)), where u(n) minimizes the functional over I-n(C). We give an implementable characterization of I-n(C). Then the finite-dimens ional problem turns out to be a minimization problem with linear constraint s. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsev ier SAS.