Exploiting cone approximations in an augmented Lagrangian method for conic optimization

TL;DR

This paper introduces a novel algorithm for nonlinear conic programming that uses cone approximations to improve convergence and efficiency, demonstrated through numerical experiments on copositive cone constraints.

Contribution

The paper presents a new augmented Lagrangian method leveraging cone approximations, ensuring strong convergence and better performance than traditional polyhedral approximation methods.

Findings

Algorithm guarantees strong global convergence to KKT points.

Polyhedral cone approximation strategy outperforms standard methods.

Numerical experiments validate improved efficiency in copositive cone problems.

Abstract

We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point is necessarily found when a limit point satisfies Robinson's condition. We conduct numerical experiments minimizing nonlinear functions subject to a copositive cone constraint. In order to do this, we consider a well known polyhedral approximation of this cone by means of refining the polyhedral constraints after each augmented Lagrangian iteration. We show that our strategy outperforms the standard approach of considering a close polyhedral approximation of the full copositive cone in every iteration.