  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.
