Skip to main content
SpringerLink
Log in

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this two-part study, we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite-dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e., whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can be performed with the use of optimality conditions and constraint qualifications. In the first paper, we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle, we recover existing simple, necessary, and sufficient conditions for the global exactness of linear penalty functions and for the existence of augmented Lagrange multipliers of Rockafellar–Wets’ augmented Lagrangian. We also present completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems as well.

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

Similar content being viewed by others

References

  1. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia (1990)

    Book  MATH  Google Scholar 

  2. Auslender, A.: Penalty and barrier methods: a unified framework. SIAM J. Optim. 10, 211–230 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birgin, E.G., Martinez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM, Philadelphia (2014)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  5. 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 

  6. 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 

  7. 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 

  8. 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 

  9. 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 

  10. Rubinov, A.M., Yang, X.: Lagrange-Type Functions in Constrained Non-convex Optimization. Kluwer Academic Publishers, Dordrecht (2003)

    Book  MATH  Google Scholar 

  11. Wang, C.Y., Yang, X.Q., Yang, X.M.: Unified nonlinear Lagrangian approach to duality and optimal paths. J. Optim. Theory Appl. 135, 85–100 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giannessi, F.: Constrained Optimization and Image Space Analysis. Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)

    MATH  Google Scholar 

  13. Li, J., Feng, S., Zhang, Z.: A unified approach for constrained extremum problems: image space analysis. J. Optim. Theory Appl. 159, 69–92 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part I: image space analysis. J. Optim. Theory Appl. 161, 738–762 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part II: special duality schemes. J. Optim. Theory Appl. 161, 763–782 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control. Optim. 27, 1333–1360 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yevtushenko, YuG, Zhadan, V.G.: Exact auxiliary functions in optimization problems. U.S.S.R. Comput. Math. Math. Phys. 30, 31–42 (1990)

    Article  MathSciNet  Google Scholar 

  18. Di Pillo, G.: Exact penalty methods. In: Spedicato, E. (ed.) Algorithms for Continuous Optimization: The State of the Art, pp. 1–45. Kluwer Academic Press, Boston (1994)

    Google Scholar 

  19. Eremin, I.I.: Penalty method in convex programming. Soviet Math. Dokl. 8, 459–462 (1966)

    MATH  Google Scholar 

  20. Zangwill, W.I.: Nonlinear programming via penalty functions. Manag. Sci. 13, 344–358 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  21. Han, S.P., Mangasarian, O.L.: Exact penalty functions in nonlinear programming. Math. Program. 17, 251–269 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Di Pillo, G., Grippo, L.: On the exactness of a class of nondifferentiable penalty functions. J. Optim. Theory Appl. 57, 399–410 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Burke, J.V.: An exact penalization viewpoint on constrained optimization. SIAM J. Control. Optim. 29, 968–998 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Demyanov, V.F.: Nonsmooth optimization. In: Di Pillo, G., Schoen, F. (eds.) Nonlinear Optimization. Lecture Notes in Mathematics, vol. 1989, pp. 55–164. Springer, Berlin (2010)

    Chapter  Google Scholar 

  25. Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization 65, 1167–1202 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fletcher, R.: A class of methods for nonlinear programming with termination and convergence properties. In: Abadie, J. (ed.) Integer and Nonlinear Programming, pp. 157–173. North-Holland, Amsterdam (1970)

    Google Scholar 

  27. Fletcher, R.: An exact penalty function for nonlinear programming with inequalities. Math. Program. 5, 129–150 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Di Pillo, G., Grippo, L.: A continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints. SIAM J. Control Optim. 23, 72–84 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lucidi, S.: New results on a continuously differentiable exact penalty function. SIAM J. Optim. 2, 558–574 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fukuda, E.H., Silva, P.J.S., Fukushima, M.: Differentiable exact penalty functions for nonlinear second-order cone programs. SIAM J. Optim. 22, 1607–1633 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Di Pillo, G., Grippo, L.: A new class of augmented Lagrangians in nonlinear programming. SIAM J. Control Optim. 17, 618–628 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Di Pillo, G., Grippo, L.: A new augmented Lagrangian function for inequality constraints in nonlinear programming problems. J. Optim. Theory Appl. 36, 495–519 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  33. Di Pillo, G., Lucidi, S.: An augmented Lagrangian function with improved exactness properties. SIAM J. Optim. 12, 376–406 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fukuda, E.H., Lourenco, B.F.: Exact augmented Lagrangian functions for nonlinear semidefinite programming. arXiv:1705.06551 (2017)

  35. Dolgopolik, M.V.: Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property. J. Glob. Optim. (2018). https://doi.org/10.1007/s10898-017-0603-0

    Google Scholar 

  36. Rubinov, A.M., Glover, B.M., Yang, X.Q.: Decreasing functions with applications to penalization. SIAM J. Optim. 10, 289–313 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rubinov, A.M., Yang, X.Q., Bagirov, A.M.: Penalty functions with a small penalty parameter. Optim. Methods Softw. 17, 931–964 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rubinov, A.M., Gasimov, R.N.: Strictly increasing positively homogeneous functions with applications to exact penalization. Optimization 52, 1–28 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, X.Q., Huang, X.X.: Partially strictly monotone and nonlinear penalty functions for constrained mathematical programs. Comput. Optim. Appl. 25, 293–311 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Huyer, W., Neumaier, A.: A new exact penalty function. SIAM J. Optim. 13, 1141–1158 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, C., Ma, C., Zhou, J.: A new class of exact penalty functions and penalty algorithms. J. Glob. Optim. 58, 51–73 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions II: parametric penalty functions. Optimization 66, 1577–1622 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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 

  44. 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 

  45. Dolgopolik, M.V.: Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle. Math. Program. 166, 297–326 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. Dolgopolik, M.V.: A unified approach to the global exactness of penalty and augmented Lagrangian functions I: parametric exactness. arXiv:1709.07073 (2017)

  47. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  48. Liu, Y.J., Zhang, L.W.: Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. J. Optim. Theory Appl. 139, 557–575 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sun, D., Sun, J., Zhang, L.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114, 349–391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wu, H., Luo, H., Ding, X., Chen, G.: Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming. Comput. Optim. Appl. 56, 531–558 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to Professor Franco Giannessi for pointing out to the author the importance of development of a general theory of the exactness of penalty and augmented Lagrangian functions several years ago. The author also wishes to express his thanks to Professor X. Yang for thoughtful and stimulating comments that helped to improve the quality of the article.

Author information

Authors and Affiliations

  1. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg State University, Saint Petersburg, Russia

    M. V. Dolgopolik

Authors
  1. M. V. Dolgopolik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Dolgopolik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dolgopolik, M.V. A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness. J Optim Theory Appl 176, 728–744 (2018). https://doi.org/10.1007/s10957-018-1238-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1238-0

Keywords

Mathematics Subject Classification

Search

Navigation