Error Stability Properties of Generalized Gradient-Type Algorithms

We present a unified framework for convergence analysis of generalized subgradient-type algorithms in the presence of perturbations. A principal novel feature of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an ɛ-stationary set of the problem, where ɛ depends on the magnitude of perturbations. Characterization of the attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp, and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel, and heavy ball modifications.