Skip to main content
Log in

Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near global optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, F.S., Wu, Z.Y., Zhu, D.L.: Lower order calmness and exact penalty function. Optim. Methods Softw. 21, 515–525 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New Jersey (2006)

    Book  MATH  Google Scholar 

  4. Birgin, E.G., Floudas, C.A., Martinez, J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Prog. Ser. A. 125, 139–162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birgin, E.G., Martinez, J.M., Prudente, L.F.: Optimality properties of an augmented Lagrangian method on infeasible problems. Comput. Optim. Appl. 60, 609–631 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Burachik, R.S., Iusem, A.N., Melo, J.G.: The exact penalty map for nonsmooth and nonconvex optimization. Optimization 64, 717–738 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burachik, R.S., Rubinov, A.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413–436 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chatzipanagiotis, N., Dentcheva, D., Zavlanos, M.M.: An augmented Lagrangian method for distributed optimization. Math. Prog. Ser. A. 152, 405–434 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Curtis, F.E., Jiang, H., Robinson, D.P.: An adaptive augmented Lagrangian method for large-scale constrained optimization. Math. Prog. Ser. A. 152, 201–245 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization 65, 1167–1202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flores-Bazán, F., Mastroeni, G.: Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications. SIAM J. Optim. 25, 647–676 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Glob. Optim. 24, 187–203 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gasimov, R.N., Rubinov, A.M.: On augmented Lagrangian for optimization problems with a single constraint. J. Glob. Optim. 28, 153–173 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, X.X., Yang, X.Q.: Further study on augmented Lagrangian duality theory. J. Glob. Optim. 31, 193–210 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hiriart-Urruty, J.-B., Strodiot, J.-J., Hien Nguyen, V.: Generalized Hessian matrix and second-order optimality conditions for problems with \(C^{1, 1}\) data. Appl. Math. Optim. 11, 43–56 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kan, C., Song, W.: Augmented Lagrangian duality for composite optimization problems. J. Optim. Theory Appl. 165, 763–784 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kan, C., Song, W.: Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems. J. Glob. Optim. 63, 77–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kelley, J.L.: General Topology. Springer, New York (1975)

    MATH  Google Scholar 

  23. Klatte, D., Tammer, K.: On second-order sufficient optimality conditions for \(C^{1, 1}\)-optimization problems. Optimization 19, 169–179 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, Q., Tang, W.M., Yang, X.M.: Properties of saddle points for generalized augmented Lagrangian. Math. Methods Oper. Res. 69, 111–124 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, Q., Yang, X.: Zero duality and saddle points of a class of augmented Lagrangian functions in constrained non-convex optimization. Optimization 57, 655–667 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meng, K.W., Yang, X.Q.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20, 3208–3231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Penot, J.-P.: Augmented Lagrangians, duality and growth conditions. J. Nonlinear Convex Anal. 3, 283–302 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, London (1969)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)

    Book  MATH  Google Scholar 

  33. Rückmann, J.J., Shapiro, A.: Augmented Lagrangians in semi-infinite programming. Math. Prog. Ser. B. 116, 499–512 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Prog. Ser. B. 77, 301–320 (1997)

    MathSciNet  MATH  Google Scholar 

  35. Shapiro, A., Sun, J.: Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29, 479–491 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, C., Liu, Q., Qu, B.: Global saddle points of nonlinear augmented Lagrangian functions. J. Glob. Optim. (2016). doi:10.1007/s10898-016-0456-y

    MATH  Google Scholar 

  38. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, Z.Y., Bai, F.S., Yang, X.Q., Zhang, L.S.: An exact lower order penalty function and its smoothing in nonlinear programming. Optimization 53, 51–68 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, H.X., Luo, H.Z., Yang, J.F.: Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming. J. Glob. Optim. 59, 695–727 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, X.Q., Meng, Z.Q.: Lagrange multipliers and calmness conditions of order \(p\). Math. Oper. Res. 32, 95–101 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhou, J., Chen, J.-S.: On the existence of saddle points for nonlinear second-order cone programming problems. J. Glob. Optim. 62, 459–480 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhou, J., Xiu, N., Wang, C.: Saddle point and exact penalty representation for generalized proximal Lagrangians. J. Glob. Optim. 56, 669–687 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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 

  46. Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian function, non-quadratic growth condition and exact penalization. Oper. Res. Lett. 34, 127–134 (2006)

    Article  MathSciNet  Google Scholar 

  47. Zhang, L., Yang, X.: An augmented Lagrangian approach with a variable transformation in nonlinear programming. Nonlinear Anal. 69, 2095–2113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian functions for constrained optimization problems. J. Glob. Optim. 52, 95–108 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhou, Y.Y., Zhou, J.C., Yang, X.Q.: Existence of augmented Lagrange multipliers for cone constrained optimization problems. J. Glob. Optim. 58, 243–260 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the anonymous referees for thoughtful and stimulating comments that helped to improve the quality of the article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. V. Dolgopolik.

Additional information

The reported study was supported by Russian Foundation for Basic Research, research Project No. 16–31–00056.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dolgopolik, M.V. Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle. Math. Program. 166, 297–326 (2017). https://doi.org/10.1007/s10107-017-1122-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-017-1122-y

Keywords

Mathematics Subject Classification

Navigation