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On a primal-dual Newton proximal method for convex quadratic programs

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  • Alberto Marchi

    (Universität der Bundeswehr München)

Abstract

This paper introduces QPDO, a primal-dual method for convex quadratic programs which builds upon and weaves together the proximal point algorithm and a damped semismooth Newton method. The outer proximal regularization yields a numerically stable method, and we interpret the proximal operator as the unconstrained minimization of the primal-dual proximal augmented Lagrangian function. This allows the inner Newton scheme to exploit sparse symmetric linear solvers and multi-rank factorization updates. Moreover, the linear systems are always solvable independently from the problem data and exact linesearch can be performed. The proposed method can handle degenerate problems, provides a mechanism for infeasibility detection, and can exploit warm starting, while requiring only convexity. We present details of our open-source C implementation and report on numerical results against state-of-the-art solvers. QPDO proves to be a simple, robust, and efficient numerical method for convex quadratic programming.

Suggested Citation

  • Alberto Marchi, 2022. "On a primal-dual Newton proximal method for convex quadratic programs," Computational Optimization and Applications, Springer, vol. 81(2), pages 369-395, March.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:2:d:10.1007_s10589-021-00342-y
    DOI: 10.1007/s10589-021-00342-y
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    References listed on IDEAS

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    1. Liqun Qi & Houyuan Jiang, 1997. "Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations," Mathematics of Operations Research, INFORMS, vol. 22(2), pages 301-325, May.
    2. Goran Banjac & Paul Goulart & Bartolomeo Stellato & Stephen Boyd, 2019. "Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 490-519, November.
    3. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    4. Philip Gill & Daniel Robinson, 2012. "A primal-dual augmented Lagrangian," Computational Optimization and Applications, Springer, vol. 51(1), pages 1-25, January.
    5. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    6. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    7. Brendan O’Donoghue & Eric Chu & Neal Parikh & Stephen Boyd, 2016. "Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1042-1068, June.
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    Cited by:

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