Mathematical Programming

, Volume 168, Issue 1–2, pp 177–200 | Cite as

Optimality conditions for nonlinear semidefinite programming via squared slack variables

  • Bruno F. Lourenço
  • Ellen H. Fukuda
  • Masao Fukushima
Full Length Paper Series B


In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of “no-gap” second-order optimality conditions that are essentially equivalent to the ones already considered in the literature. We conclude with the analysis of some computational prospects of the squared slack variables approach for NSDP.


Nonlinear semidefinite programming Squared slack variables Optimality conditions Second-order conditions 

Mathematics Subject Classification

90C22 90C46 90C30 


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Bruno F. Lourenço
    • 1
  • Ellen H. Fukuda
    • 2
  • Masao Fukushima
    • 3
  1. 1.Department of Computer and Information Science, Faculty of Science and TechnologySeikei UniversityMusashino-shiJapan
  2. 2.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
  3. 3.Department of Systems and Mathematical Science, Faculty of Science and EngineeringNanzan UniversityNagoyaJapan

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