An algorithm for finding a first-order and second-order critical point of composite nonsmooth problems is proposed in this paper. For smooth problems, algorithms for searching such a point usually utilize the so called negative-curvature directions. In this paper, the method recently proposed for nonlinear semidefinite problems by the current author is extended for solving general composite nonsmooth problems. Acceleration by Newton-like method is also proposed, where near a solution, the active functions are identified, and solving a set of linear equations is suffice to give the local quadratic convergence. It is also shown that by further solving another set of linear equations, the second-order correction is possible to avoid the Maratos effect.
NTT DATA Mathematical Systems Inc. Technical Report 35 Shinanomachi Shinjuku Tokyo Japan June/2022