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On Accuracy Estimates for One Regularization Method in Linear Programming

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Optimization Problems and Their Applications (OPTA 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 871))

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Abstract

In this paper, the alternative duality schemes for mathematical programming problems are considered. These schemes are based on the Lagrange function regularized by Tikhonov in primal and dual variables simultaneously. Earlier such schemes were investigated for convex programming problems only, and the conditions were found which guarantee a convergence of both primal and dual components of the saddle point of the regularized Lagrange function to the optimal sets of primal and dual problems respectively. In addition, some accuracy estimates were obtained for deviation of only one of these components from the normal solution (solution with minimal the Euclidean norm) of the corresponding problem, primal or dual. Unfortunately, these estimates are valid only if primal and dual parameters of regularization have a different order of smallness. In this article, the linear case is investigated in detail. It is shown that for linear programming problem both mentioned sequences converge to the normal solutions of primal and dual programs simultaneously, and it is not essential for such a convergence whether the regularization parameters have a different or the same order of smallness. Also, the alternative accuracy estimates are presented which appear to be more precise and efficient in comparison with the estimates known for a convex case.

This work was supported by Russian Science Foundation, grant N 14–11–00109.

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Notes

  1. 1.

    It follows from well-known Hoffman’s lemma [11] estimating the distance of an arbitrary point to the polyhedral set defined by the system of linear inequalities (and the optimal set of a linear programming problem belongs to such a class) in terms of deviations of this point from each inequality separately.

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Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics UB RAS, Ekaterinburg, Russia

    Leonid D. Popov

  2. Ural Federal University, Ekaterinburg, Russia

    Leonid D. Popov

Authors
  1. Leonid D. Popov
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Correspondence to Leonid D. Popov .

Editor information

Editors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, Omsk, Russia

    Anton Eremeev

  2. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  3. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Yury Kochetov

  4. Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos Pardalos

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Popov, L.D. (2018). On Accuracy Estimates for One Regularization Method in Linear Programming. In: Eremeev, A., Khachay, M., Kochetov, Y., Pardalos, P. (eds) Optimization Problems and Their Applications. OPTA 2018. Communications in Computer and Information Science, vol 871. Springer, Cham. https://doi.org/10.1007/978-3-319-93800-4_14

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  • DOI: https://doi.org/10.1007/978-3-319-93800-4_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93799-1

  • Online ISBN: 978-3-319-93800-4

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