qpOASES: a parametric active-set algorithm for quadratic programming

Abstract

Many practical applications lead to optimization problems that can either be stated as quadratic programming (QP) problems or require the solution of QP problems on a lower algorithmic level. One relatively recent approach to solve QP problems are parametric active-set methods that are based on tracing the solution along a linear homotopy between a QP problem with known solution and the QP problem to be solved. This approach seems to make them particularly suited for applications where a-priori information can be used to speed-up the QP solution or where high solution accuracy is required. In this paper we describe the open-source C++ software package qpOASES, which implements a parametric active-set method in a reliable and efficient way. Numerical tests show that qpOASES can outperform other popular academic and commercial QP solvers on small- to medium-scale convex test examples of the Maros-Mészáros QP collection. Moreover, various interfaces to third-party software packages make it easy to use, even on embedded computer hardware. Finally, we describe how qpOASES can be used to compute critical points of nonconvex QP problems.

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Notes

  1. 1.

    The name qpOASES is derived from the term “online active set strategy” [19] reflecting the fact that the code has been originally developed for use in model predictive control applications.

  2. 2.

    We use the notion warm-start if the QP solution procedure is initialized based on the solution of the previous QP problem. Hot-start refers to the case where also internal matrix factorizations are re-used.

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Acknowledgments

The authors would like to thank the three anonymous referees whose insightful comments have helped to improve this article. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement no  FP7-ICT-2009-4 248940 (EMBOCON). The Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp, DFG GSC 220/2) supported this work by providing travel grants. Moreover, this research was supported by the German Research Association (DFG) under grants BO 864/12, BO 864/13, and 864/15, and the Research Council KUL: PFV/10/002 Optimization in Engineering Center OPTEC, GOA/10/09 MaNet and GOA/10/11 Global real-time optimal control of autonomous robots and mechatronic systems. Furthermore, this research was supported by the Flemish Government: IOF / KP / SCORES4CHEM; by FWO: PhD/postdoc grants, projects G.0320.08 (convex MPC) and G.0377.09 (Mechatronics MPC); by IWT: PhD Grants, SBO LeCoPro project; by the Belgian Federal Science Policy Office: IUAP P7 (DYSCO, Dynamical systems, control and optimization, 2012-2017); and by the EU: FP7-SADCO (MC ITN-264735), FP7-TEMPO (MC ITN-607957), ERC ST HIGHWIND (259 166), Eurostars SMART, ACCM. Development of a first prototype version of qpOASES has been supported by the REGINS-PREDIMOT European project. At the time of initial submission, the first author was with the Eletrical Engineering Department of KU Leuven (Belgium) and held a PhD fellowship of the Research Foundation – Flanders (FWO).

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Correspondence to Hans Joachim Ferreau.

Appendix

Appendix

See Table 5.

Table 5 Comparison of QP solvers on problems from the Maros-Mészáros test set ([44]) with at most \(n = 1{,}000\) variables and \(m=1{,}001\) two-sided inequality constraints (not counting variable bound constraints). The solution time in seconds and the optimality condition residual \(\rho \) are given for each solver. Failure of the solver is denoted by blank time fields

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Ferreau, H.J., Kirches, C., Potschka, A. et al. qpOASES: a parametric active-set algorithm for quadratic programming. Math. Prog. Comp. 6, 327–363 (2014). https://doi.org/10.1007/s12532-014-0071-1

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Keywords

  • Parametric quadratic programming
  • Active set method
  • Model predictive control

Mathematics Subject Classification

  • 90C20 (Quadratic programming)
  • 65K05 (Mathematical programming methods)