General Proximal Quasi-Newton Methods based on model functions for nonsmooth nonconvex problems

TL;DR

This paper introduces a flexible proximal quasi-Newton method for nonconvex, nonsmooth optimization that does not require bounded variable metrics, demonstrating convergence and effectiveness through numerical experiments.

Contribution

It develops a novel proximal quasi-Newton algorithm with unbounded variable metrics, extending existing methods for nonconvex nonsmooth problems.

Findings

Sequences converge to stationary points under mild conditions.

Variable metric boundedness depends on objective regularity.

Numerical results outperform proximal gradient methods.

Abstract

In this manuscript, we propose a general proximal quasi-Newton method tailored for nonconvex and nonsmooth optimization problems, where we do not require the sequence of the variable metric (or Hessian approximation) to be uniformly bounded as a prerequisite, instead, the variable metric is updated by a continuous matrix generator. From the respective of the algorithm, the objective function is approximated by the so-called local model function and subproblems aim to exploit the proximal point(s) of such model function, which help to achieve the sufficiently decreasing functional sequence along with the backtracking line search principle. Under mild assumptions in terms of the first-order information of the model function, every accumulation point of the generated sequence is stationary and the sequence of the variable metric is proved not to be bounded. Additionally, if the function has the Kurdyka-{\L}ojasiewicz property at the corresponding accumulation point, we find that the whole sequence is convergent to the stationary point, and the sequence of the variable metric is proved to be uniformly bounded. Through the above results, we think that the boundedness of the sequence of the variable metric should depend on the regularity of objectives, rather than being assumed as a prior for nonsmooth optimization problems. Numerical experiments on polytope feasibility problems and (sparse) quadratic inverse problems demonstrate the effectiveness of our proposed model-based proximal quasi-Newton method, in comparison with the associated model-based proximal gradient method.