Optimality conditions for nonlinear semidefinite programming via squared slack variables

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

Abstract

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.

Keywords

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

Mathematics Subject Classification

90C22 90C46 90C30 

References

  1. 1.
    Barvinok, A.: Problems of distance geometry and convex properties of quadratic maps. Discrete Comput. Geom. 13(1), 189–202 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bernstein, D.S.: Matrix Mathematics: Theory, Facts, and Formulas, 2nd edn. Princeton University Press, Princeton (2009)Google Scholar
  3. 3.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific (1999)Google Scholar
  4. 4.
    Bonnans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9(2), 466–492 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burer, S., Monteiro, R.D.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Progr. 95(2), 329–357 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Burer, S., Monteiro, R.D.: Local minima and convergence in low-rank semidefinite programming. Math. Progr. 103(3), 427–444 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21(1), 265–287 (1990)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fiala, J., Kočvara, M., Stingl, M.: PENLAB: a matlab solver for nonlinear semidefinite optimization. ArXiv e-prints (2013)Google Scholar
  9. 9.
    Forsgren, A.: Optimality conditions for nonconvex semidefinite programming. Math. Progr. 88(1), 105–128 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fukuda, E.H., Fukushima, M.: The use of squared slack variables in nonlinear second-order cone programming. J. Optim. Theory Appl. (2016). doi:10.1007/s10957-016-0904-3
  11. 11.
    Hock, W., Schittkowski, K.: Test examples for nonlinear programming codes. J. Optim. Theory Appl. 30(1), 127–129 (1980)CrossRefMATHGoogle Scholar
  12. 12.
    Jarre, F.: Elementary optimality conditions for nonlinear SDPs. In: Handbook on Semidefinite, Conic and Polynomial Optimization, International Series in Operations Research and Management Science (vol. 166, pp. 455–470). Springer, Berlin (2012)Google Scholar
  13. 13.
    Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Progr. 41(1–3), 73–96 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kočvara, M., Stingl, M.: PENNON: a code for convex nonlinear and semidefinite programming. Optim. Methods Softw. 18(3), 317–333 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 1st edn. Springer, New York (1999)CrossRefMATHGoogle Scholar
  16. 16.
    Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23(2), 339–358 (1998)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pataki, G.: The geometry of semidefinite programming. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  18. 18.
    Robinson, S.M.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497–513 (1976)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Schittkowski, K.: Test examples for nonlinear programming codes—all problems from the Hock-Schittkowski-collection. Department of Computer Science, University of Bayreuth, Tech. rep. (2009)Google Scholar
  20. 20.
    Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Progr. 77(1), 301–320 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sturm, J.F.: Similarity and other spectral relations for symmetric cones. Linear Algebra Appl. 312(1–3), 135–154 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yamashita, H., Yabe, H.: A survey of numerical methods for nonlinear semidefinite programming. J. Oper. Res. Soc. Jpn. 58(1), 24–60 (2015)MathSciNetCrossRefMATHGoogle Scholar

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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