An abstract convergence framework with application to inertial inexact forward--backward methods

TL;DR

This paper introduces a new abstract convergence framework for nonsmooth nonconvex optimization, enabling the design of inertial algorithms with convergence guarantees and demonstrated effectiveness in image deblurring tasks.

Contribution

It presents a novel abstract descent scheme that generalizes key properties for convergence and proposes two inertial algorithms with inexactness criteria, improving optimization in nonsmooth nonconvex problems.

Findings

Algorithms converge to stationary points under Kurdyka-Lojasiewicz property.

Inertial algorithms can escape local minima due to their inertial features.

Demonstrated effectiveness on image deblurring problems with linesearch techniques.

Abstract

In this paper we introduce a novel abstract descent scheme suited for the minimization of proper and lower semicontinuous functions. The proposed abstract scheme generalizes a set of properties that are crucial for the convergence of several first-order methods designed for nonsmooth nonconvex optimization problems. Such properties guarantee the convergence of the full sequence of iterates to a stationary point, if the objective function satisfies the Kurdyka-Lojasiewicz property. The abstract framework allows for the design of new algorithms. We propose two inertial-type algorithms with implementable inexactness criteria for the main iteration update step. The first algorithm, i$^2$Piano, exploits large steps by adjusting a local Lipschitz constant. The second algorithm, iPila, overcomes the main drawback of line-search based methods by enforcing a descent only on a merit function instead of the objective function. Both algorithms have the potential to escape local minimizers (or stationary points) by leveraging the inertial feature. Moreover, they are proved to enjoy the full convergence guarantees of the abstract descent scheme, which is the best we can expect in such a general nonsmooth nonconvex optimization setup using first-order methods. The efficiency of the proposed algorithms is demonstrated on two exemplary image deblurring problems, where we can appreciate the benefits of performing a linesearch along the descent direction inside an inertial scheme.