  In this paper we theoretically show that interior-point methods based on
self-concordant barriers possess favorable global complexity beyond their
standard application area of convex optimization. To do that we propose first-
and second-order methods for non-convex optimization problems with general
convex set constraints and linear constraints. Our methods attain a suitably
defined class of approximate first- or second-order KKT points with the
worst-case iteration complexity similar to unconstrained problems, namely
$O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order),
respectively.
