A gradient method with inexact oracle for composite nonconvex optimization

In this paper, we develop new first-order method for composite non-convex minimization problems with simple constraints and inexact oracle. The objective function is given as a sum of `hard', possibly part, `simple' convex part. Informally speaking, oracle inexactness means that, the `hard' at any point can approximately calculate value construct quadratic function, which bounds from above. We give several examples such inexactness: smooth functions Holder-continuous gradient, by auxiliary uniformly concave maximization problem, be solved only approximately. For introduced class problems, propose gradient-type method, allows to use different proximal setup adapt geometry feasible set, adaptively chooses controlled error, mapping. provide convergence rate our in terms norm generalized gradient mapping show case universal respect Holder parameters problem. Finally, particular case, that small necessary condition local minimum holds point.