TY - JOUR

T1 - QPALM

T2 - a proximal augmented lagrangian method for nonconvex quadratic programs

AU - Hermans, Ben

AU - Themelis, Andreas

AU - Patrinos, Panagiotis

N1 - Funding Information:
The work of Panagiotis Patrinos was supported by the Research Foundation Flanders (FWO) research projects G081222N, G086518N, G086318N, and G0A0920N; Research Council KU Leuven C1 project No. C14/18/068; Fonds de la Recherche Scientifique—FNRS and the Fonds Wetenschappelijk Onderzoek—Vlaanderen under EOS project no 30468160 (SeLMA); European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 953348.
Funding Information:
The work of Ben Hermans was supported by KU Leuven-BOF PFV/10/002 Centre of Excellence: Optimization in Engineering (OPTEC), from project G0C4515N of the Research Foundation–Flanders (FWO–Flanders), from Flanders Make ICON: Avoidance of collisions and obstacles in narrow lanes, and from the KU Leuven Research project C14/15/067: B-spline based certificates of positivity with applications in engineering.
Funding Information:
The work of Andreas Themelis was supported by the JSPS KAKENHI grant number JP21K17710.
Publisher Copyright:
© 2022, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85126867941&partnerID=8YFLogxK

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U2 - 10.1007/s12532-022-00218-0

DO - 10.1007/s12532-022-00218-0

M3 - Article

AN - SCOPUS:85126867941

JO - Mathematical Programming Computation

JF - Mathematical Programming Computation

SN - 1867-2949

ER -