On the Iterate Convergence and Manifold Identification of Inexact Proximal-Newton-Type Methods Under a Sharpness Condition

We consider minimizing the sum of a smooth function and a regularization term that promotes structured solution using an inexact proximal variable-metric approach, which encompasses widely-used algorithms including proximal Newton and proximal quasi-Newton. Convergence of the iterates is proven when both terms are convex and the sum satisfies a sharpness condition locally, of which the quadratic growth condition and the weak sharp minima are special cases. When the regularization is partly smooth, a condition satisfied by many popular subproblem solvers for obtaining approximate update steps is then presented to be sufficient for identification of the optimal manifold around a local solution. Through the theoretical results, IPVM+, an algorithm with improved time efficiency, is then presented in detail with numerical results on real-world problems.