Adaptive, anisotropic and hierarchical cones of discrete convex functions

Abstract

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 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.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):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.We refer to the numerical section §6, and to [24] for the economic model details; let us only say here that the Monopolist's optimal product line is {∇U (z); z ∈ Ω}, and that the optimal prices are given by the Legendre-Fenchel dual of U . Consider the following three regions, defined for k ∈ {0, 1, 2} (implicitly excluding points z ∈ Ω close to which U is not smooth) Ω k := {z ∈ Ω; rank(Hessian U (z)) = k}.(2) *

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