QPALM: A Newton-type Proximal Augmented Lagrangian Method for Quadratic Programs

TL;DR

QPALM is an efficient Newton-type solver for convex quadratic programs that combines proximal augmented Lagrangian methods with semismooth Newton iterations, offering fast convergence and effective infeasibility detection.

Contribution

The paper introduces QPALM, a novel solver that integrates proximal augmented Lagrangian techniques with semismooth Newton methods for improved quadratic programming performance.

Findings

Outperforms state-of-the-art QP solvers on various benchmarks.

Efficient line search reduces to a simple zero-finding problem.

Can enforce primal and dual residuals to strict tolerances.

Abstract

We present a proximal augmented Lagrangian based solver for general convex quadratic programs (QPs), relying on semismooth Newton iterations with exact line search to solve the inner subproblems. The exact line search reduces in this case to finding the zero of a one-dimensional monotone, piecewise affine function and can be carried out very efficiently. Our algorithm requires the solution of a linear system at every iteration, but as the matrix to be factorized depends on the active constraints, efficient sparse factorization updates can be employed like in active-set methods. Both primal and dual residuals can be enforced down to strict tolerances and otherwise infeasibility can be detected from intermediate iterates. A C implementation of the proposed algorithm is tested and benchmarked against other state-of-the-art QP solvers for a large variety of problem data and shown to compare favorably against these solvers.