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.

翻译：对于一些典型和广泛使用的非convex半赤道半赤道正规化模型和Ambrosio-Tortorelli近似Mumford-Shah模型,我们根据Kurdyka-L Ojasiewicz分析以及最近的非convex近效算法,开发了一个高效的前提条件框架,目的是解决非线性交替最小化程序中出现的线性子问题。解决大尺度线性子问题对于许多交替最小化算法来说总是重要和具有挑战性。我们通过将高效和传统的迭代与非线性和非线性最小化优化合作,证明线性子问题只需要一个或任何限定的固定时间,而不将错误控制为通常的不精确解答器。拟议的前提条件框架可以为处理线性子问题提供极大的灵活性和效率,并保证非线性交替最小化方法的全球趋同。