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.

Nonconvex Generalization of ADMM for Nonlinear Equality Constrained Problems

The ever-increasing demand for efficient and distributed optimization algorithms for large-scale data has led to the growing popularity of the Alternating Direction Method of Multipliers (ADMM). However, although the use of ADMM to solve linear equality constrained problems is well understood, we lacks a generic framework for solving problems with nonlinear equality constraints, which are common in practical applications (e.g., spherical constraints). To address this problem, we are proposing a new generic ADMM framework for handling nonlinear equality constraints, neADMM. After introducing the generalized problem formulation and the neADMM algorithm, the convergence properties of neADMM are discussed, along with its sublinear convergence rate $o(1/k)$, where $k$ is the number of iterations. Next, two important applications of neADMM are considered and the paper concludes by describing extensive experiments on several synthetic and real-world datasets to demonstrate the convergence and effectiveness of neADMM compared to existing state-of-the-art methods

Computing equilibria of Cournot oligopoly models with mixed-integer quantities

We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called “2-groups partitionable” Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method