An Active Set Algorithm for Nonlinear Optimization with Polyhedral Constraints

TL;DR

This paper introduces PASA, an active set algorithm for nonlinear optimization with polyhedral constraints, combining gradient projection and linearly constrained optimization methods, with proven convergence properties.

Contribution

It presents a novel active set algorithm that seamlessly integrates two phases and provides convergence guarantees under specific conditions.

Findings

Global convergence to stationary points

Asymptotic phase two performance under certain conditions

Effective handling of polyhedral constraints in nonlinear optimization

Abstract

A polyhedral active set algorithm PASA is developed for solving a nonlinear optimization problem whose feasible set is a polyhedron. Phase one of the algorithm is the gradient projection method, while phase two is any algorithm for solving a linearly constrained optimization problem. Rules are provided for branching between the two phases. Global convergence to a stationary point is established, while asymptotically PASA performs only phase two when either a nondegeneracy assumption holds, or the active constraints are linearly independent and a strong second-order sufficient optimality condition holds.