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Quasi-Newton approaches to interior point methods for quadratic problems

Author

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  • J. Gondzio

    (University of Edinburgh)

  • F. N. C. Sobral

    (State University of Maringá)

Abstract

Interior point methods (IPM) rely on the Newton method for solving systems of nonlinear equations. Solving the linear systems which arise from this approach is the most computationally expensive task of an interior point iteration. If, due to problem’s inner structure, there are special techniques for efficiently solving linear systems, IPMs demonstrate a reduced computing time and are able to solve large scale optimization problems. It is tempting to try to replace the Newton method by quasi-Newton methods. Quasi-Newton approaches to IPMs either are built to approximate the Lagrangian function for nonlinear programming problems or provide an inexpensive preconditioner. In this work we study the impact of using quasi-Newton methods applied directly to the nonlinear system of equations for general quadratic programming problems. The cost of each iteration can be compared to the cost of computing correctors in a usual interior point iteration. Numerical experiments show that the new approach is able to reduce the overall number of matrix factorizations.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:1:d:10.1007_s10589-019-00102-z
    DOI: 10.1007/s10589-019-00102-z
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    References listed on IDEAS

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    1. Gondzio, Jacek, 1995. "HOPDM (version 2.12) -- A fast LP solver based on a primal-dual interior point method," European Journal of Operational Research, Elsevier, vol. 85(1), pages 221-225, August.
    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. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    4. Mauricio G. C. Resende & K. G. Ramakrishnan & Zvi Drezner, 1995. "Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming," Operations Research, INFORMS, vol. 43(5), pages 781-791, October.
    5. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.
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    4. Kirschner, Felix, 2023. "Conic optimization with applications in finance and approximation theory," Other publications TiSEM e9bef4a5-ee46-45be-90d7-9, Tilburg University, School of Economics and Management.

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