Interior Point Method with Modified Augmented Lagrangian for Penalty-Barrier Nonlinear Programming

TL;DR

This paper introduces a novel numerical method combining a modified augmented Lagrangian approach with interior-point techniques to efficiently solve nonlinear programming problems with large penalties and poorly scaled barriers.

Contribution

It presents a new solver that directly minimizes a merit function using specialized techniques for quadratic penalties and logarithmic barriers, with proven convergence and polynomial complexity in linear cases.

Findings

Global convergence and local quadratic convergence are established.

The method achieves weak polynomial time complexity for linear programs.

The approach effectively handles large penalties and poorly scaled barriers.

Abstract

We present a numerical method for the local solution of nonlinear programming problems. The SUMT approach of Fiacco and McCormick results in a merit function with quadratic penalties and logarithmic barriers. Our NLP solver works by directly minimizing this merit function. In our method, we use different concepts that each shall aim at the efficient treatment of one respective special feature of this merit function. The features are: large quadratic penalty terms, and badly scaled logarithmic barriers. The quadratic penalties are treated with a modified Augmented Lagrangian technique. It enables large step sizes despite nonlinearity of the equality constraints. The logarithmic barriers we treat with a primal-dual interior-point path-following technique. We prove global convergence of the method and local quadratic convergence. We further prove weak polynomial time-complexity in the special case where the nonlinear program is a linear program. We also use a trust-funnel so to avoid that the method converges to any stationary points that are infeasible to the constraints.