  We analyse the convergence of the gradient projection algorithm, which is
finalized with the Newton method, to a stationary point for the problem of
nonconvex constrained optimization $\min_{x \in S} f(x)$ with a proximally
smooth set $S = \{x \in R^n : g(x) = 0 \}, \; g : R^n \rightarrow R^m$ and a
smooth function $f$. We propose new Error bound (EB) conditions for the
gradient projection method which lead to the convergence domain of the Newton
method. We prove that these EB conditions are typical for a wide class of
optimization problems. It is possible to reach high convergence rate of the
algorithm by switching to the Newton method.
