A proximal minimization algorithm for structured nonconvex and nonsmooth problems

TL;DR

This paper introduces a full splitting proximal algorithm designed for complex structured nonconvex and nonsmooth optimization problems, ensuring convergence to critical points under certain conditions.

Contribution

It presents a novel proximal splitting method tailored for structured nonconvex, nonsmooth problems with convergence guarantees and rates based on the Kurdyka-ojasiewicz property.

Findings

Algorithm converges to KKT points under mild conditions.

Global convergence established with convergence rates.

Applicable to problems with multiple nonsmooth and smooth components.

Abstract

We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function, each of the nonsmooth summands depending on an independent block variable, and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the boundedness of the generated sequence and prove that any cluster point of the latter is a KKT point of the minimization problem. In the setting of the Kurdyka-\L{}ojasiewicz property we show global convergence, and derive convergence rates for the iterates in terms of the \L{}ojasiewicz exponent.