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New Perspective on Slack Variables Applications to Singular Optimization Problems

  • Yuri Evtushenko
  • Vlasta Malkova
  • Alexey Tret’yakovEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (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.

Keywords

Degeneracy Nonlinear programming p-factor method Superlinear convergence 2-regularity 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dorodnicyn Computing Centre, FRC CSC RASMoscowRussia
  2. 2.Faculty of SciencesSiedlce UniversitySiedlcePoland
  3. 3.System Research InstitutePolish Academy of SciencesWarsawPoland
  4. 4.Moscow Institute of Physics and Technology (State University)Dolgoprudny, Moscow RegionRussia

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