Adaptive, anisotropic and hierarchical cones of discrete convex functions

• Published in: Numerische Mathematik Vol. 132; no. 4; pp. 807 - 853
• Main Author: Mirebeau, Jean-Marie
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2016; Springer Verlag
• Subjects: Analysis of PDEs; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Simulation; Theoretical; 65K15; 52A41; 26B25; 90C25; 91B24; 65N06; Convex optimization; Monopolist problem; Global convexity constraint
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-015-0732-7

We introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions. Among the various mathematical problems which possess such a formulation, the Monopolist problem (Rochet and Choné, Econometrica 66:783–826, 1998 ; Ekeland and Moreno-Bromberg, Numer Math 115:45–69, 2010 ) arising in economics is our main motivation. Consider a two dimensional domain Ω , sampled on a grid X of N points. We show that the cone Conv ( X ) of restrictions to X of convex functions on Ω is typically 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 Conv ( V ) of Conv ( X ) , associated to stencils V which can be adaptively, locally, and anisotropically refined. We show, using the arithmetic structure of the grid, that the trace U | X of any convex function U on Ω is contained in a cone Conv ( V ) defined by only O ( N ln 2 N ) linear constraints, in average over grid orientations. Numerical experiments for the Monopolist problem, based on adaptive stencil refinement strategies, show that the proposed method offers an unrivaled accuracy/complexity trade-off in comparison with existing methods. We also obtain, as a side product of our theory, a new average complexity result on edge flipping based mesh generation.