A Newton-Type Active Set Method for Nonlinear Optimization with Polyhedral Constraints

TL;DR

This paper introduces a Newton-type active set method for large-scale nonlinear optimization with polyhedral constraints, combining gradient projection and Newton steps, achieving quadratic convergence under certain conditions.

Contribution

The paper proposes a novel Newton-type active set algorithm that effectively handles large-scale problems with polyhedral constraints, demonstrating quadratic convergence.

Findings

Algorithm asymptotically takes Newton steps under certain conditions.

Method exhibits quadratic convergence rate.

Numerical experiments show strong performance on benchmark problems.

Abstract

A Newton-type active set algorithm for large-scale minimization subject to polyhedral constraints is proposed. The algorithm consists of a gradient projection step, a second-order Newton-type step in the null space of the constraint matrix, and a set of rules for branching between the two steps. We show that the proposed method asymptotically takes the Newton step when the active constraints are linearly independent and a strong second-order sufficient optimality condition holds. We also show that the method has a quadratic rate of convergence under standard conditions. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEst and on a specific class of problems for which finding second-order stationary points is critical.