QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs

TL;DR

QPALM is a new solver for nonconvex quadratic programs that uses a proximal augmented Lagrangian approach, ensuring convergence to stationary points efficiently and robustly across various problem types.

Contribution

It introduces QPALM, a novel algorithm combining proximal augmented Lagrangian methods with semismooth Newton steps for nonconvex QPs, with open-source implementation and strong empirical performance.

Findings

Successfully solves all Maros-Meszaros problems

Finds stationary points for most nonconvex QPs in tests

Competitive performance against state-of-the-art solvers

Abstract

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an \(R\)-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.