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International Conference on Optimization and Applications

OPTIMA 2018: Optimization and Applications pp 3–20Cite as

New Perspective on Slack Variables Applications to Singular Optimization Problems

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 974))

Abstract

This paper is devoted to a new approach for solving nonlinear programming (NLP) problems for which the Kuhn-Tucker optimality conditions system of equations is singular. It happens when the strict complementarity condition (SCC), a constrained qualification (CQ), and a second-order sufficient condition (SOSC) for optimality is not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equality. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient optimality condition.

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research (projects no. 17-07-00510, 17-07-00493) and the RAS Presidium Program (program 27).

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Authors and Affiliations

  1. Dorodnicyn Computing Centre, FRC CSC RAS, Vavilov st. 40, 119333, Moscow, Russia

    Yuri Evtushenko, Vlasta Malkova & Alexey Tret’yakov

  2. Faculty of Sciences, Siedlce University, 08-110, Siedlce, Poland

    Alexey Tret’yakov

  3. System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland

    Alexey Tret’yakov

  4. Moscow Institute of Physics and Technology (State University), 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia

    Yuri Evtushenko

Authors
  1. Yuri Evtushenko
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  2. Vlasta Malkova
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  3. Alexey Tret’yakov
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Corresponding author

Correspondence to Alexey Tret’yakov .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Yury Evtushenko

  2. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  3. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  4. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Yury Kochetov

  5. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Mikhail Posypkin

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Evtushenko, Y., Malkova, V., Tret’yakov, A. (2019). New Perspective on Slack Variables Applications to Singular Optimization Problems. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_1

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  • DOI: https://doi.org/10.1007/978-3-030-10934-9_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10933-2

  • Online ISBN: 978-3-030-10934-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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