  Interior point methods (IPMs) that handle nonconvex constraints such as
IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in
the setting where the objective and constraints are thrice differentiable, and
have Lipschitz first and second derivatives on the feasible region. We provide
an IPM that, starting from a strictly feasible point, finds a $\mu$-approximate
Fritz John point by solving $\mathcal{O}( \mu^{-7/4})$ trust-region
subproblems. For IPMs that handle nonlinear constraints, this result represents
the first iteration bound with a polynomial dependence on $1/\mu$. We also show
how to use our method to find scaled-KKT points starting from an infeasible
solution and improve on existing complexity bounds.
