A Preconditioned Alternating Minimization Framework for Nonconvex and Half Quadratic Optimization

TL;DR

This paper introduces a preconditioned alternating minimization framework for nonconvex and half-quadratic optimization problems, improving efficiency and convergence in solving large-scale linear subproblems within nonlinear algorithms.

Contribution

It develops a novel preconditioned framework that efficiently solves linear subproblems in nonconvex alternating minimization, ensuring global convergence without requiring inexact error control.

Findings

Only finite preconditioned iterations needed per subproblem

Framework guarantees global convergence

Enhanced efficiency for large-scale problems

Abstract

For some typical and widely used non-convex half-quadratic regularization models and the Ambrosio-Tortorelli approximate Mumford-Shah model, based on the Kurdyka-\L ojasiewicz analysis and the recent nonconvex proximal algorithms, we developed an efficient preconditioned framework aiming at the linear subproblems that appeared in the nonlinear alternating minimization procedure. Solving large-scale linear subproblems is always important and challenging for lots of alternating minimization algorithms. By cooperating the efficient and classical preconditioned iterations into the nonlinear and nonconvex optimization, we prove that only one or any finite times preconditioned iterations are needed for the linear subproblems without controlling the error as the usual inexact solvers. The proposed preconditioned framework can provide great flexibility and efficiency for dealing with linear subproblems and guarantee the global convergence of the nonlinear alternating minimization method simultaneously.