7,053 research outputs found
Handling convexity-like constraints in variational problems
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
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.Comment: 23 page
Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions
We address the discretization of optimization problems posed on the cone of
convex functions, motivated in particular by the principal agent problem in
economics, which models the impact of monopoly on product quality. Consider a
two dimensional domain, sampled on a grid of N points. We show that the cone of
restrictions to the grid of convex functions is in general characterized by N^2
linear inequalities; a direct computational use of this description therefore
has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of
discrete convex functions, associated to stencils which can be adaptively,
locally, and anisotropically refined. Numerical experiments optimize the
accuracy/complexity tradeoff through the use of a-posteriori stencil refinement
strategies.Comment: 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2.
Modifications are anecdotic.
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