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Primal-Dual Relationship Between Levenberg–Marquardt and Central Trajectories for Linearly Constrained Convex Optimization

Author

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  • Roger Behling

    (Católica SC)

  • Clovis Gonzaga

    (Federal University of Santa Catarina)

  • Gabriel Haeser

    (Federal University of São Paulo)

Abstract

We consider the minimization of a convex function on a bounded polyhedron (polytope) represented by linear equality constraints and non-negative variables. We define the Levenberg–Marquardt and central trajectories starting at the analytic center using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of the parameter. Based on this, we develop an algorithm that starts computing primal-dual feasible points on the Levenberg–Marquardt trajectory and eventually moves to the central path. Our main theorem is particularly relevant in quadratic programming, where points on the primal-dual Levenberg–Marquardt trajectory can be calculated by means of a system of linear equations. We present some computational tests related to box constrained trust region subproblems.

Suggested Citation

  • Roger Behling & Clovis Gonzaga & Gabriel Haeser, 2014. "Primal-Dual Relationship Between Levenberg–Marquardt and Central Trajectories for Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 705-717, September.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:3:d:10.1007_s10957-013-0492-4
    DOI: 10.1007/s10957-013-0492-4
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    References listed on IDEAS

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    1. M. D. Gonzalez-Lima & C. Roos, 2005. "On Central-Path Proximity Measures in Interior-Point Methods," Journal of Optimization Theory and Applications, Springer, vol. 127(2), pages 303-328, November.
    2. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    3. L. M. Graña Drummond & B. F. Svaiter, 1999. "On Well Definedness of the Central Path," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 223-237, August.
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    Cited by:

    1. Baha Alzalg, 2019. "A primal-dual interior-point method based on various selections of displacement step for symmetric optimization," Computational Optimization and Applications, Springer, vol. 72(2), pages 363-390, March.

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