Skip to main content
SpringerLink
Log in

Global saddle points of nonlinear augmented Lagrangian functions

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We notice that the results for the existence of global (local) saddle points of augmented Lagrangian functions in the literature were only sufficient conditions of some special types of augmented Lagrangian. In this paper, we introduce a general class of nonlinear augmented Lagrangian functions for constrained optimization problem. In two different cases, we present sufficient and necessary conditions for the existence of global saddle points. Moreover, as corollaries of the two results above, we not only obtain sufficient and necessary conditions for the existence of global saddle points of some special types of augmented Lagrangian functions mentioned in the literature, but also give some weaker sufficient conditions than the ones in the literature. Compared with our recent work (Wang et al. in Math Oper Res 38:740–760, 2013), the nonlinear augmented Lagrangian functions in this paper are more general and the results in this paper are original. We show that some examples (such as improved barrier augmented Lagrangian) satisfy the assumptions of this paper, but not available in Wang et al. (2013).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)

    MATH  Google Scholar 

  3. Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  4. Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 524–532 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, T., Wang, Y.J., Liang, Z.A., Pardalos, P.M.: Local saddle point and a class of convexification methods for noncovex optimization problems. J. Glob. Optim. 38, 405–419 (2007)

    Article  MATH  Google Scholar 

  6. Liu, Q., Yang, X.M., Lee, H.W.: On saddle points of a class of augmented Lagrangian functions. J. Ind. Manag. Optim. 3, 693–700 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rockafellar, R.T.: Lagrange multipliers and optimization. SIAM Rev. 35, 183–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rockafellar, R.T., Wets, J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  11. Sun, X.L., Li, D., McKinnon, K.I.M.: On saddle points of augmented Lagrangians for constrained nonconvex optimization. SIAM J. Optim. 15, 1128–1146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang, C.Y., Li, D.: Unified theory of augmented Lagrangian methods for constrained global optimization. J. Glob. Optim. 44, 433–458 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear augmented Lagrangian and duality theory. Math. Oper. Res. 38, 740–760 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, C.Y., Zhou, J.C., Xu, X.H.: Saddle point theory of two classes of augmented Lagrangian and its applications to generalized semi-infinite programming. Appl. Math. Optim. 59, 413–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu, H.X., Luo, H.Z.: Saddle points of general augmented Lagrangians for constrained nonconvex optimization. J. Glob. Optim. 53, 683–697 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhou, J.C., Xiu, N.H., Wang, C.Y.: Saddle point and exact penalty representation for generalized proximal Lagrangians. J. Glob. Optim. 54, 669–687 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhou, Y.Y., Yang, X.Q.: Some results about duality and exact penalization. J. Glob. Optim. 29, 497–509 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the two anonymous referees for their valuable comments and constructive suggestions for improving the paper.

Author information

Authors and Affiliations

  1. Institute of Operations Research, Qufu Normal University, Qufu, 273165, China

    Changyu Wang

  2. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China

    Qian Liu

  3. School of Management, Qufu Normal University, Rizhao, 276826, China

    Biao Qu

Authors
  1. Changyu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Biao Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qian Liu.

Additional information

This research was partially supported by the National Natural Science Foundation of China (10971118, 10901096, 11271226, and 11271233) and the Natural Science Foundation of Shandong Province (ZR2013FL032).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Liu, Q. & Qu, B. Global saddle points of nonlinear augmented Lagrangian functions. J Glob Optim 68, 125–146 (2017). https://doi.org/10.1007/s10898-016-0456-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-016-0456-y

Keywords

Search

Navigation