QPALM
swMATH ID:  35397 
Software Authors:  Ben Hermans, Andreas Themelis, Panagiotis Patrinos 
Description:  QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs. 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 of a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extendedrealvalued cost function. Furthermore, we prove global convergence of such iterations to a stationary point at an Rlinear rate in the specific case of a (possibly nonconvex) QP. 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, piecewise affine function. QPALM is implemented in opensource C code, with tailored linear algebra routines for the factorization in a selfwritten package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the MarosMeszaros 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 stateoftheart 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. 
Homepage:  https://arxiv.org/abs/2010.02653 
Keywords:  Optimization; arXiv_math.OC; nonconvex quadratic programming; QP solver; proximal augmented Lagrangian method; C code 
Related Software:  OSQP; qpOASES; NETLIB LP Test Set; Mosek; CHOLMOD; Gurobi; FBstab; Matlab; LDL; ALGENCAN; Ipopt; GALAHAD; HPIPM; CUTEst; BPMPD; AMD; OOQP; GitHub; NASOQ; qpDUNES 
Cited in:  7 Publications 
Standard Articles
1 Publication describing the Software  Year 

QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs Ben Hermans, Andreas Themelis, Panagiotis Patrinos 
2020

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Cited by 23 Authors
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Cited in 6 Serials
2  Mathematical Programming Computation 
1  Automatica 
1  Applied Numerical Mathematics 
1  SIAM Journal on Optimization 
1  Computational Optimization and Applications 
1  Optimization Letters 
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