Felipe Atenas

A proximal subgradient method for nonconvex stochastic optimization under the Kurdyka-Łojasiewicz condition

Categories Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , , , ,

This work introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function. We target a broad class of integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, a structural assumption that simultaneously covers … Read more

A relaxed version of Ryu’s three-operator splitting method for structured nonconvex optimization

Categories Nonlinear Optimization, Nonsmooth Optimization

In this work, we propose a modification of Ryu’s splitting algorithm for minimizing the sum of three functions, where two of them are convex with Lipschitz continuous gradients, and the third is an arbitrary proper closed function that is not necessarily convex. The modification is essential to facilitate the convergence analysis, particularly in establishing a … Read more

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

Categories Convex and Nonsmooth Optimization, Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , ,

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

Splitting Methods for Nonconvex Optimisation: Convergence and Saddle Point Avoidance Through Shadow Sequences

Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , , , ,

In nonconvex optimisation, commonly used methods are usually shown to converge to stationary points, while guarantees of convergence to local minimisers have comparatively received less attention. To close this gap, we analyse the convergence behaviour of the Davis–Yin three-operator splitting method. More precisely, for structured weakly convex semialgebraic optimisation problems, we establish convergence to local … Read more

Weak convexity and approximate subdifferentials

Categories Complementarity and Variational Inequalities, Generalized Convexity/Monoticity, Nonsmooth Optimization

We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new … Read more

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

Categories Complementarity and Variational Inequalities, Nonlinear Optimization Tags , , , , , , ,

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