Marc Teboulle In this paper, we aim at unifying, simplifying, and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central role in the unification and in the simplification of the analysis of all Lagrangian-based methods. Equipped with a nice primal … Read more
We consider a nonconvex optimization problem consisting of maximizing the difference of two convex functions. We present a randomized method that requires low computational effort at each iteration. The described method is a randomized coordinate descent method employed on the so-called Toland-dual problem. We prove subsequence convergence to dual stationarity points, a new notion that … Read more
This paper introduces a method for computing points satisfying the second-order necessary optimality conditions in constrained nonconvex minimization. The method comprises two independent steps corresponding to the first and second order conditions. The first-order step is a generic closed map algorithm which can be chosen from a variety of first-order algorithms, making it The second-order … Read more
We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding smoothness restriction is pervasive in first order methods (FOM), and was recently circumvent for convex composite optimization by Bauschke, Bolte and Teboulle, … Read more
We introduce a class of nonconvex/affine feasibility problems, called (NCF), that consists of finding a point in the intersection of affine constraints with a nonconvex closed set. This class captures some interesting fundamental and NP hard problems arising in various application areas such as sparse recovery of signals and affine rank minimization that we briefly … Read more
We consider the problem of locating a single radiating source from several noisy measurements using a maximum likelihood (ML) criteria. The resulting optimization problem is nonconvex and nonsmooth and thus finding its global solution is in principal a hard task. Exploiting the special structure of the objective function, we introduce and analyze two iterative schemes … Read more