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

TL;DR

This paper establishes polynomial iteration bounds for interior point methods solving nonconvex constrained problems, providing new theoretical guarantees and practical algorithms for finding approximate Fritz John and KKT points.

Contribution

It introduces an interior point method with polynomial iteration bounds for nonconvex constraints, a first in this setting, and extends to infeasible starting points.

Findings

IPMs find a $$-approximate Fritz John point in $\u00a0()^{-7/4}$ trust-region subproblems

First polynomial iteration bound for IPMs with nonlinear constraints

Improved complexity bounds for finding scaled-KKT points from infeasible solutions

Abstract

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.