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Crash start of interior point methods

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  • Gondzio, Jacek

Abstract

The starting point used by an interior point algorithm for linear and convex quadratic programming may significantly influence the behaviour of the method. A widely used heuristic to construct such a point consists of dropping variable nonnegativity constraints and computing a solution which minimizes the Euclidean norm of the primal (or dual) point while satisfying the appropriate primal (or dual) equality constraints, followed by shifting the variables so that all their components are positive and bounded away from zero. In this Short Communication a new approach for finding a starting point is proposed. It relies on a few inexact Newton steps performed at the start of the solution process. A formal justification of the new heuristic is given and computational results are presented to demonstrate its advantages in practice. Computational experience with a large collection of small- and medium-size test problems reveals that the new starting point is superior to the old one and saves 20–40% of iterations needed by the primal-dual method. For larger and more difficult problems this translates into remarkable savings in the solution time.

Suggested Citation

  • Gondzio, Jacek, 2016. "Crash start of interior point methods," European Journal of Operational Research, Elsevier, vol. 255(1), pages 308-314.
    • Handle: RePEc:eee:ejores:v:255:y:2016:i:1:p:308-314
    DOI: 10.1016/j.ejor.2016.05.030
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    References listed on IDEAS

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    1. J. Hall, 2010. "Towards a practical parallelisation of the simplex method," Computational Management Science, Springer, vol. 7(2), pages 139-170, April.
    2. Robert E. Bixby, 1992. "Implementing the Simplex Method: The Initial Basis," INFORMS Journal on Computing, INFORMS, vol. 4(3), pages 267-284, August.
    3. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
    4. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.
    5. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
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