Handling Convexity-Like Constraints in Variational Problems

Abstract

Abstract. We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms, and can be easily parallelized, thus making it applicable to large scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation : 3D denoising, the principal agent problem, and optimization within the class of convex bodies.

Discretization of functionals involving the Monge–Ampère operator

“…Oberman used the idea of imposing convexity constraints on a wide-stencils [24], which amounts to only selecting the constraints that involve nearby points in the formulation of [11]. Oudet and Mérigot [21] used interpolation operators to approximate the solutions of (1.7) on more general finite-dimensional spaces of functions. All these methods can be used to minimize functionals that involve the value of the function and its gradient only.…”

{Euclidean, metric, and Wasserstein} gradient flows: an overview

“…Because of this difficulty, a different approach is needed. For instance, Ekeland and Moreno-Bromberg used the representation of a convex function as a maximum of affine functions [40], but this needed many more linear constraints; Oudet and Mérigot [75] decided to test convexity on a different (and less refined) grid than that where the functions are defined… These methods give somehow satisfactory answers for functionals involving u and ∇u, but are not able to handle terms involving the Monge-Ampère operator det (D 2 u).…”

Numerical resolution of Euler equations through semi-discrete optimal transport

“…First, the constraint of convexity can be discretized in various ways, none of which is particularly simple or canonical [2,34,35]. For the problem of interest, convexity can also be imposed through the discretization of the Monge-Ampere operator [26,8].…”