Adaptive, anisotropic and hierarchical cones of discrete convex functions

We introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions. Among the various mathematical problems which posses such a formulation, the Monopolist problem [24, 10] 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 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 lnN) 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. A number of mathematical problems can be formulated as the optimization of a convex functional over the cone of convex functions on a domain Ω (here compact and two dimensional): Conv(Ω) := {U : Ω→ R; U is convex}. This includes optimal transport, as well as various geometrical conjectures such as Newton’s problem [16, 18]. We choose for concreteness to emphasize an economic application: the Monopolist (or Principal Agent) problem [24], in which the objective is to design an optimal product line, and an optimal pricing catalog, so as to maximize profit in a captive market. The following minimal instance is numerically studied in [1, 10, 21] and on Figure 1. With Ω = [1, 2]2