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An interior point-proximal method of multipliers for convex quadratic programming

Author

Listed:
  • Spyridon Pougkakiotis

    (University of Edinburgh)

  • Jacek Gondzio

    (University of Edinburgh)

Abstract

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

Suggested Citation

  • Spyridon Pougkakiotis & Jacek Gondzio, 2021. "An interior point-proximal method of multipliers for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 78(2), pages 307-351, March.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:2:d:10.1007_s10589-020-00240-9
    DOI: 10.1007/s10589-020-00240-9
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    References listed on IDEAS

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    1. Spyridon Pougkakiotis & Jacek Gondzio, 2019. "Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 905-945, June.
    2. Philip Gill & Daniel Robinson, 2012. "A primal-dual augmented Lagrangian," Computational Optimization and Applications, Springer, vol. 51(1), pages 1-25, January.
    3. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
    4. 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.
    5. Paul Armand & Ngoc Nguyen Tran, 2019. "An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 197-215, April.
    6. Paul Armand & Riadh Omheni, 2017. "A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 523-547, May.
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    Cited by:

    1. Silvia Berra & Alessandro Torraca & Federico Benvenuto & Sara Sommariva, 2024. "Combined Newton-Gradient Method for Constrained Root-Finding in Chemical Reaction Networks," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 404-427, January.
    2. Jacek Gondzio & Spyridon Pougkakiotis & John W. Pearson, 2022. "General-purpose preconditioning for regularized interior point methods," Computational Optimization and Applications, Springer, vol. 83(3), pages 727-757, December.
    3. Stefano Cipolla & Jacek Gondzio, 2023. "Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1061-1103, June.
    4. Spyridon Pougkakiotis & Jacek Gondzio, 2022. "An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 97-129, January.
    5. Xin Jiang & Lieven Vandenberghe, 2022. "Bregman primal–dual first-order method and application to sparse semidefinite programming," Computational Optimization and Applications, Springer, vol. 81(1), pages 127-159, January.

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