Understanding the Douglas-Rachford splitting method through the lenses of Moreau-type envelopes

We analyze the Douglas-Rachford splitting method for weakly convex optimization problems, by the token of the Douglas-Rachford envelope, a merit function akin to the Moreau envelope. First, we use epi-convergence techniques to show that this artifact approximates the original objective function via epigraphs. Secondly, we present how global convergence and local linear convergence rates for … Read more

Weakly convex Douglas-Rachford splitting avoids strict saddle points

We prove that the Douglas-Rachford splitting method converges, almost surely, to local minimizers of semialgebraic weakly convex optimization problems, under the assumption of the strict saddle property. The approach consists of two steps: first, we prove a manifold identification result, and local smoothness of the involved iteration operator. Then, we proceed to show that strict … Read more

A unified analysis of descent sequences in weakly convex optimization, including convergence rates for bundle methods

We present a framework for analyzing convergence and local rates of convergence of a class of descent algorithms, assuming the objective function is weakly convex. The framework is general, in the sense that it combines the possibility of explicit iterations (based on the gradient or a subgradient at the current iterate), implicit iterations (using a … Read more

Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, … Read more

Composite optimization for robust blind deconvolution

The blind deconvolution problem seeks to recover a pair of vectors from a set of rank one bilinear measurements. We consider a natural nonsmooth formulation of the problem and show that under standard statistical assumptions, its moduli of weak convexity, sharpness, and Lipschitz continuity are all dimension independent. This phenomenon persists even when up to … Read more

On proximal point-type algorithms for weakly convex functions and their connection to the backward Euler method

In this article we study the connection between proximal point methods for nonconvex optimization and the backward Euler method from numerical Ordinary Differential Equations (ODEs). We establish conditions for which these methods are equivalent. In the case of weakly convex functions, for small enough parameters, the implicit steps can be solved using a strongly convex … Read more