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, delu) on the set C of convex func tions u in an appropriate functional space X, Such problems arise for insta nce in mathematical economics [4]. A special case gives the convex envelope u(0)** of a given function u(0). Let (T-n) be any quasiuniform sequence of meshes whose diameter goes to zero, and I-n the corresponding affine inter polation operators. We prove that the minimizer over C is the limit of the sequence (u(n)), where u(n) minimizes the functional over I-n(C). We give a n implementable characterization of I-n(C). Then the finite dimensional pro blem turns out to be a minimization problem with linear constraints.