Splitting methods with variable metric for KL functions

TL;DR

This paper analyzes the convergence of variable metric splitting methods for nonconvex functions satisfying the Kurdyka-Lojasiewicz property, providing new convergence rates and covering advanced algorithm variants.

Contribution

It introduces convergence analysis for variable metric splitting methods applied to nonconvex functions with Kurdyka-Lojasiewicz property, including a nonsmooth Levenberg-Marquardt variant.

Findings

Sequences converge to critical points under broad conditions

New convergence rates for function values and iterates

Analysis includes advanced algorithm variants like variable metric and errors

Abstract

We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satisfies the Kurdyka-Lojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of f and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailled.