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A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

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  • David Ek

    (KTH Royal Institute of Technology)

  • Anders Forsgren

    (KTH Royal Institute of Technology)

Abstract

The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions. We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

Suggested Citation

  • David Ek & Anders Forsgren, 2023. "A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming," Computational Optimization and Applications, Springer, vol. 86(1), pages 1-48, September.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:1:d:10.1007_s10589-023-00486-z
    DOI: 10.1007/s10589-023-00486-z
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    References listed on IDEAS

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    1. Paul Armand & Joël Benoist & Jean-Pierre Dussault, 2012. "Local path-following property of inexact interior methods in nonlinear programming," Computational Optimization and Applications, Springer, vol. 52(1), pages 209-238, May.
    2. Benedetta Morini & Valeria Simoncini & Mattia Tani, 2017. "A comparison of reduced and unreduced KKT systems arising from interior point methods," Computational Optimization and Applications, Springer, vol. 68(1), pages 1-27, September.
    3. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
    4. J. Gondzio & F. N. C. Sobral, 2019. "Quasi-Newton approaches to interior point methods for quadratic problems," Computational Optimization and Applications, Springer, vol. 74(1), pages 93-120, September.
    5. Benedetta Morini & Valeria Simoncini, 2017. "Stability and Accuracy of Inexact Interior Point Methods for Convex Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 450-477, November.
    6. David Ek & Anders Forsgren, 2021. "Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization," Computational Optimization and Applications, Springer, vol. 79(1), pages 155-191, May.
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