A primal-dual interior-point relaxation method with global and rapidly local convergence for nonlinear programs

TL;DR

This paper introduces a novel primal-dual interior-point relaxation method for nonlinear programming that ensures global and rapid local convergence, effectively handling ill-conditioning and jamming issues in existing methods.

Contribution

The paper proposes a new relaxation approach based on a parametric mini-max reformulation, enabling convergence without interior-point constraints and extending to general inequality constraints.

Findings

Method achieves global convergence to KKT points.

Numerical results show efficiency on challenging problems.

Potential to overcome jamming and ill-conditioning issues.

Abstract

Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs with general equality and nonnegative constraints. In each iteration, our method approximately solves the KKT system of a parametric equality constrained mini-max subproblem, which avoids the requirement that any primal or dual iterate is an interior-point. The method has some similarities to the warmstarting interior-point methods in relaxing the interior-point requirement and is easily extended for solving problems with general inequality constraints. In particular, it has the potential to circumvent the jamming difficulty that appears with many interior-point methods for nonlinear programs and improve the ill conditioning of existing primal-dual interior-point methods as the barrier parameter is small. A new smoothing approach is introduced to develop our relaxation method and promote convergence of the method. Under suitable conditions, it is proved that our method can be globally convergent and locally quadratically convergent to the KKT point of the original problem. The preliminary numerical results on a well-posed problem for which many interior-point methods fail to find the minimizer and a set of test problems from the CUTEr collection show that our method is efficient.