An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians

  • Regina S. Burachik
  • Alfredo N. Iusem
  • Jefferson G. Melo


We consider a primal optimization problem in a reflexive Banach space and a duality scheme via generalized augmented Lagrangians. For solving the dual problem (in a Hilbert space), we introduce and analyze a new parameterized Inexact Modified Subgradient (IMSg) algorithm. The IMSg generates a primal-dual sequence, and we focus on two simple new choices of the stepsize. We prove that every weak accumulation point of the primal sequence is a primal solution and the dual sequence converges weakly to a dual solution, as long as the dual optimal set is nonempty. Moreover, we establish primal convergence even when the dual optimal set is empty. Our second choice of the stepsize gives rise to a variant of IMSg which has finite termination.


Banach spaces Nonsmooth optimization Nonconvex optimization Duality scheme Augmented Lagrangian Inexact modified subgradient algorithm 



Regina S. Burachik acknowledges support by the Australian Research Council Discovery Project Grant DP0556685 for this study; Jefferson G. Melo acknowledges support from CNPq, through a scholarship for a one-year long visit to University of South Australia.

We thank the reviewers for their careful reading and comments.


  1. 1.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  2. 2.
    Penot, J.P.: Augmented Lagrangians, duality and growth conditions. J. Nonlinear Convex Anal. 3, 283–302 (2002) MathSciNetMATHGoogle Scholar
  3. 3.
    Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Huang, X.X., Yang, X.Q.: Further study on augmented Lagrangian duality theory. J. Glob. Optim. 31, 193–210 (2005) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian function, non-quadratic growth condition and exact penalization. Oper. Res. Lett. 34, 127–134 (2006) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burachik, R.S., Rubinov, A.M.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413–436 (2007) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Zhou, Y.Y., Yang, X.Q.: Duality and penalization in optimization via an augmented Lagrangian function with applications. J. Optim. Theory Appl. 140, 171–188 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Burachik, R.S., Iusem, A.N., Melo, J.G.: Duality and exact penalization for general augmented Lagrangians. J. Optim. Theory Appl. 147, 125–140 (2010) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Nedić, A., Ozdaglar, A.: Approximate primal solutions and rate analysis for dual subgradient methods. SIAM J. Optim. 19, 1757–1780 (2009) MATHCrossRefGoogle Scholar
  10. 10.
    Nedić, A., Ozdaglar, A.: Subgradient methods for saddle-point problems. J. Optim. Theory Appl. 142, 205–228 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Glob. Optim. 24, 187–203 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Burachik, R.S., Gasimov, R.N., Ismayilova, N.A., Kaya, C.Y.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Glob. Optim. 34, 55–78 (2006) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Burachik, R.S., Kaya, C.Y., Mammadov, M.: An inexact modified subgradient algorithm for nonconvex optimization. Comput. Optim. Appl. 45, 1–24 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Burachik, R.S., Iusem, A.N., Melo, J.G.: A primal dual modified subgradient algorithm with sharp Lagrangian. J. Glob. Optim. 46, 347–361 (2010) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Burachik, R.S., Kaya, C.Y.: An update rule and a convergence result for a penalty function method. J. Ind. Manag. Optim. 3, 381–398 (2007) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41, 591–616 (2000) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Volkwein, S.: Mesh-independence for an augmented Lagrangian-SQP method in Hilbert spaces. SIAM J. Control Optim. 38, 767–785 (2000) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Gasimov, R.N., Rubinov, A.M.: On augmented Lagrangians for optimization problems with a single constraint. J. Glob. Optim. 28, 153–173 (2004) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Burachik, R.S., Rubinov, A.M.: On the absence of duality gap for Lagrange-type functions. J. Ind. Manag. Optim. 1, 33–38 (2005) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Rubinov, A.M., Yang, X.Q.: Lagrange-type Functions in Constrained Non-convex Optimization. Kluwer Academic, Amsterdam (2003) MATHGoogle Scholar
  21. 21.
    Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2008) Google Scholar
  22. 22.
    Alber, Ya.I., Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–35 (1998) MathSciNetMATHGoogle Scholar
  23. 23.
    Burachik, R.S., Kaya, C.Y.: A deflected subgradient method using a general augmented Lagrangian duality with implications on penalty methods. In: Burachik, R.S., Yao, J.C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control, vol. 47, pp. 109–132. Springer, New York (2010) CrossRefGoogle Scholar
  24. 24.
    Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Springer, Berlin (1981) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Regina S. Burachik
    • 1
  • Alfredo N. Iusem
    • 2
  • Jefferson G. Melo
    • 3
  1. 1.School of Mathematics and StatisticsUniversity of South AustraliaMawson LakesAustralia
  2. 2.IMPAInstituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil
  3. 3.IME/UFG, Instituto de Matemática e EstatísticaUniversidade Federal de GoiásGoiâniaBrazil

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