On the asymptotic behavior of a higher-order extrapolation primal-dual interior-point method for nonlinear programming

TL;DR

This paper analyzes a higher-order extrapolation primal-dual interior-point method for nonlinear programming, demonstrating asymptotic convergence properties and accelerating convergence using derivative-based extrapolation with numerical validation.

Contribution

It introduces a higher-order derivative framework for extrapolation in interior-point methods, improving convergence analysis and providing numerical acceleration techniques.

Findings

Asymptotic convergence is achieved with high-order derivative extrapolation.

Extrapolation accelerates convergence in quadratic programming examples.

The method extends primal-dual interior-point techniques with higher-order information.

Abstract

A trajectory-following primal--dual interior-point method solves nonlinear optimization problems with inequality and equality constraints by approximately finding points satisfying perturbed Karush--Kuhn--Tucker optimality conditions for a decreasing order of perturbation controlled by the barrier parameter. Under some conditions, there is a unique local correspondence between small residuals of the optimality conditions and points yielding that residual, and the solution on the barrier trajectory for the next barrier parameter can be approximated using an approximate solution for the current parameter. A framework using higher-order derivative information of the correspondence is analyzed in which an extrapolation step to the trajectory is first taken after each decrease of the barrier parameter upon reaching a sufficient approximation. It suffices asymptotically to only take extrapolation steps for convergence at the rate the barrier parameter decreases with when using derivative information of high enough order. Numerical results for quadratic programming problems are presented using extrapolation as accelerator.