We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational problems, and partial differential equation techniques. The approach is to approximate the (non-polyhedral) cone of convex functions by a polyhedral cone which can be represented by linear inequalities. This approach leads to an optimization problem with linear constraints which can be computed efficiently, hundreds of times faster than existing methods.
We describe an algorithm to approximate the minimizer of an elliptic functional in the form ( 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 (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 ( C ) is the limit of the 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 dimensional problem turns out to be a minimization problem with linear constraints.  Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in Hk(Ω), and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.