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.

翻译：我们为Hilbert空间的复合问题和可能的非混凝土最小化问题开发了全球化的Proximal Newton方法。此外,我们对复合目标功能的假设比现有理论的限制性要小一些,考虑到不同性和共性。关于目标功能的顺利部分的可区分性,我们引入了二阶半悬浮的概念,并讨论了它为什么构成我们Proximal Newton方法的适当框架。然而,全球趋同和地方加速都仍然存在于我们的假设中。最终,我们算法的趋同性通过解决功能空间中的玩具模型问题而表现出来。