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.

翻译：内点法（IPMs）在处理非凸约束时，如IPOPT、KNITRO和LOQO等算法，已在实践中取得巨大成功。我们考虑目标函数和约束均为三阶可微，且其可行域上的一阶和二阶导数满足Lipschitz连续性的内点法情景。提出一种内点法，从严格可行点出发，通过求解$\mathcal{O}( \mu^{-7/4})$个置信域子问题，找到$\mu$-近似Fritz John点。对于处理非线性约束的内点法，该结果首次给出了与$1/\mu$呈多项式依赖关系的迭代界。我们还展示了如何使用该方法从不可行解出发找到尺度化KKT点，并改进了现有复杂度界。