Malitsky-Tam forward-reflected-backward splitting method for nonconvex minimization problems

TL;DR

This paper extends the Malitsky-Tam forward-reflected-backward splitting method to nonconvex minimization problems, proving convergence to stationary points under a generalized KL property and demonstrating competitive numerical performance.

Contribution

The paper introduces a novel extension of the FRB splitting method for nonconvex problems with convergence guarantees based on the generalized concave KL property.

Findings

Global convergence to stationary points is established.

The method exhibits finite length property.

Numerical results show competitiveness with existing methods.

Abstract

We extend the Malitsky-Tam forward-reflected-backward (FRB) splitting method for inclusion problems of monotone operators to nonconvex minimization problems. By assuming the generalized concave Kurdyka-{\L}ojasiewicz (KL) property of a quadratic regularization of the objective, we show that the FRB method converges globally to a stationary point of the objective and enjoys finite length property. The sharpness of our approach is guaranteed by virtue of the exact modulus associated with the generalized concave KL property. Numerical experiments suggest that FRB is competitive compared to the Douglas-Rachford method and the Bo\c{t}-Csetnek inertial Tseng's method.