A numerical approach to variational problems subject to convexity constraint

Abstract

We describe an algorithm to approximate the minimizer of an elliptic functional in the form \(\int_\Omega j(x, u, \nabla u)\) on the set \({\cal C}\) 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 \(u_0\). Let \((T_n)\) be any quasiuniform sequence of meshes whose diameter goes to zero, and \(I_n\) the corresponding affine interpolation operators. We prove that the minimizer over \({\cal C}\) is the limit of the sequence \((u_n)\), where \(u_n\) minimizes the functional over \(I_n({\cal C})\). We give an implementable characterization of \(I_n({\cal C})\). Then the finite dimensional problem turns out to be a minimization problem with linear constraints.