This study considers a proximal Newton-type method to solve the minimization of a composite function that is the sum of a smooth nonconvex function and a nonsmooth convex function. In general, the method uses the Hessian matrix of the smooth portion of the objective function or its approximation. The uniformly positive definiteness of the matrix plays an important role in establishing the global convergence of the method. In this study, an inexact proximal memoryless quasi-Newton method is proposed based on the memoryless Broyden family with the modified spectral scaling secant condition. The proposed method inexactly solves the subproblems to calculate scaled proximal mappings. The approximation matrix is shown to retain the uniformly positive definiteness and the search direction is a descent direction. Using these properties, the proposed method is shown to have global convergence for nonconvex objective functions. Furthermore, the R-linear convergence for strongly convex objective functions is proved. Finally, some numerical results are provided.

这项研究考虑了一种近端牛顿型方法来解决复合函数的最小化问题，该复合函数是光滑的非凸函数和非光滑的凸函数之和。通常，该方法使用目标函数的平滑部分或其近似的Hessian矩阵。矩阵的一致正定性在建立方法的全局收敛性方面起着重要作用。在这项研究中，提出了一种不精确的近端无记忆准牛顿法，该方法基于无记忆的Broyden族并具有改进的光谱缩放割线条件。所提出的方法不精确地解决了子问题以计算缩放的近端映射。近似矩阵显示为保持一致的正定性，搜索方向为下降方向。使用这些属性，该方法对非凸目标函数具有全局收敛性。此外，证明了强凸目标函数的R线性收敛。最后，提供了一些数值结果。