Second order semi-smooth Proximal Newton methods in Hilbert spaces
针对希尔伯特空间中的复合非凸最小化问题，提出一种全局化的近端牛顿法，在更宽松的可微性和凸性假设下，仍能保证全局收敛和局部加速，并通过函数空间中的玩具模型验证。
Abstract We develop a globalized Proximal Newton method for composite and possibly non-convex minimization problems in Hilbert spaces. Additionally, we impose less restrictive assumptions on the composite objective functional considering differentiability and convexity than in existing theory. As far as differentiability of the smooth part of the objective function is concerned, we introduce the notion of second order semi-smoothness and discuss why it constitutes an adequate framework for our Proximal Newton method. However, both global convergence as well as local acceleration still pertain to hold in our scenario. Eventually, the convergence properties of our algorithm are displayed by solving a toy model problem in function space.