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Interior point methods (IPMs) that handle nonconvex constraints such asIPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs inthe setting where the objective and constraints are thrice differentiable, andhave Lipschitz first and second derivatives on the feasible region. We providean IPM that, starting from a strictly feasible point, finds a $\mu$-approximateFritz John point by solving $\mathcal{O}( \mu^{-7/4})$ trust-regionsubproblems. For IPMs that handle nonlinear constraints, this result representsthe first iteration bound with a polynomial dependence on $1/\mu$. We also showhow to use our method to find scaled-KKT points starting from an infeasiblesolution and improve on existing complexity bounds.

Worst-case iteration bounds for log barrier methods on problems with nonconvex constraints