Effects of Finite-Precision Arithmetic on Interior-Point Methods for Nonlinear Programming
Stephen J. Wright
TL;DR
The paper analyzes how finite-precision arithmetic affects primal-dual interior-point methods for nonlinear programming under the Mangasarian-Fromovitz constraint qualification and strict complementarity. It develops a structured error analysis for both condensed and augmented PDIP step formulations, showing that perturbations concentrate in subspaces that do not impede rapid convergence, and that centrality conditions enable progress even as ${\mu}$ approaches the unit roundoff ${\bf u}$. A key technical result bounds the exact primal-dual step length and extends superlinear convergence to nonconvex problems. The study also validates backward-stable pivoting strategies (BK/BP and Gaussian elimination) for the augmented system and demonstrates with numerical experiments that finite-precision PDIP preserves fast convergence well above ${\bf u}$-level precision, with implications for both PDIP and stabilized SQP methods.
Abstract
We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign, provided that the iterates satisfy centrality and feasibility conditions of the type usually associated with path-following methods. When we replace the standard assumption that the active constraint gradients are independent by the weaker Mangasarian-Fromovitz constraint qualification, rapid convergence usually is attainable, even when cancellation and roundoff errors occur during the calculations. In deriving our main results, we prove a key technical result about the size of the exact primal-dual step. This result can be used to modify existing analysis of primal-dual interior-point methods for convex programming, making it possible to extend the superlinear local convergence results to the nonconvex case.