A class of nonlinear Lagrangians for nonconvex second order cone programming

  • Liwei Zhang
  • Jian Gu
  • Xiantao Xiao


This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.


Second order cone optimization Augmented Lagrangian Nonlinear Lagrange method 


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  1. 1.
    Auslender, A., Cominetti, R., Haddou, M.: Asymptotic analysis of penalty and barrier methods in convex and linear programming. Math. Oper. Res. 22, 43–62 (1997) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ben-Tel, A., Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7(2), 347–366 (1997) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982) MATHGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999) MATHGoogle Scholar
  5. 5.
    Bonnans, J.F., Ramírez, C.H.: Perturbation analysis of second order cone programming problems. Math. Program., Ser. B 104, 205–227 (2005) MATHCrossRefGoogle Scholar
  6. 6.
    Breitfeld, M., Shanno, D.: A globally convergent penalty-barrier algorithm for nonlinear programming and its computational performance. Technical Report, Rutcor Research Report, Rutgers University, New Jersey (1994) Google Scholar
  7. 7.
    Charalambous, C.: Nonlinear least pth optimization and nonlinear programming. Math. Program. 12, 195–225 (1977) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dussault, J.P.: Augmented non-quadratic penalty algorithms. Math. Program. 99, 467–486 (2004) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendon, Oxford (1994) MATHGoogle Scholar
  10. 10.
    Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2001) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Griva, I., Polyak, R.A.: Primal-dual nonlinear rescaling method with dynamic scaling parameter update. Math. Program. 106, 237–259 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593–615 (2005) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hestenes, M.R.: Multiplier and gradient method. J. Optim. Theory Appl. 4, 303–320 (1969) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kato, H., Fukushima, M.: An SQP-type algorithm for nonlinear second-order cone programs. Optim. Lett. 1, 129–144 (2007) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Li, D.: Zero duality gap for a class of nonconvex optimization problems. J. Optim. Theory Appl. 85, 309–323 (1995) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Liu, Y.J., Zhang, L.W.: Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal. 67, 1359–1373 (2007) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Liu, Y.J., Zhang, L.W.: On the approximate augmented Lagrangian for nonlinear second order cone programming. Nonlinear Anal. Theory Methods Appl. (2008). doi: 10.1016/ Google Scholar
  18. 18.
    Mangasarian, O.L.: Unconstrained Lagrangians in nonlinear programming. SIAM J. Control 13, 772–791 (1975) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Polyak, R.A.: Modified barrier function: theory and methods. Math. Program. 54, 177–222 (1992) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Polyak, R.A.: Log-Sigmoid multipliers method in constrained optimization. Ann. Oper. Res. 101, 427–460 (2001) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Polyak, R.A.: Nonlinear rescaling vs. smoothing technique in convex optimization. Math. Program. 92, 197–235 (2002) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Polyak, R.A., Griva, I.: Primal-Dual nonlinear rescaling method for convex optimization. J. Optim. Theory Appl. 122, 111–156 (2004) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Polyak, R.A., Teboulle, M.: Nonlinear rescaling and proximal-like methods in convex optimization. Math. Program. 76, 265–284 (1997) MathSciNetMATHGoogle Scholar
  24. 24.
    Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969) Google Scholar
  25. 25.
    Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Program. 5, 354–373 (1973) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Rockafellar, R.T.: The multiplier method of Hestenes and Powell applied to convex programming. J. Optim. Theory Appl. 12, 555–562 (1973) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control 12, 268–285 (1974) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Sun, D.F., Sun, J.: Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33 421–445 (2008) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Sun, J., Zhang, L.W., Wu, Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. 129, 437–456 (2006) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Sun, D.F., Sun, J., Zhang, L.W.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. (2008) doi: 10.1007/s10107-007-0105-9 Google Scholar
  31. 31.
    Stingl, M.: On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods. Shaker, Aachen (2006). Dissertation Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianChina
  2. 2.School of ScienceDalian Fisheries UniversityDalianChina

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