, Volume 56, Issue 2, pp 390–408 | Cite as

A second-order convergence augmented Lagrangian method using non-quadratic penalty functions

  • M. D. Sánchez
  • M. L. SchuverdtEmail author
Theoretical Article


The purpose of the present paper is to study the global convergence of a practical Augmented Lagrangian model algorithm that considers non-quadratic Penalty–Lagrangian functions. We analyze the convergence of the model algorithm to points that satisfy the Karush–Kuhn–Tucker conditions and also the weak second-order necessary optimality condition. The generation scheme of the Penalty–Lagrangian functions includes the exponential penalty function and the logarithmic-barrier without using convex information.


Nonlinear programming Augmented Lagrangian methods Global convergence Constraint qualifications Sequential optimality conditions 



  1. 1.
    Abadie, J.: On the Kuhn–Tucker theorem. In: Abadie, J. (ed.) Nonlinear Programming, pp. 19–36. North Holland, Amsterdam (1967)Google Scholar
  2. 2.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18(4), 1286–1309 (2007)CrossRefGoogle Scholar
  3. 3.
    Andreani, R., Birgin, E., Martínez, J.M., Schuverdt, M.L.: Second-order negative-curvature methods for box-constrained and general constrained optimization. Comput. Optim. Appl. 45, 209–236 (2010)CrossRefGoogle Scholar
  4. 4.
    Andreani, R., Echagüe, C.E., Schuverdt, M.L.: Constant-rank condition and second-order constraint qualification. J. Optim. Theory. Appl 146, 155–266 (2009)Google Scholar
  5. 5.
    Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new weak constraint qualifications and applications. SIAM J. Optim. 22(3), 1109–1135 (2012)CrossRefGoogle Scholar
  6. 6.
    Andreani, R., Haeser, G., Ramos, A., Silva, P.J.S.: A second-order sequential optimality condition associated to the convergence of optimization algorithms. IMA J. Numer. Anal. 37(4), 1902–1929 (2017)CrossRefGoogle Scholar
  7. 7.
    Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26(1), 96–110 (2016)CrossRefGoogle Scholar
  8. 8.
    Andreani, R., Martínez, J.M., Schuverdt, M.L.: On second-order optimality conditions for Nonlinear Programming. Optimization 56, 529–542 (2007)CrossRefGoogle Scholar
  9. 9.
    Ben-Tal, A., Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7, 347–366 (1997)CrossRefGoogle Scholar
  10. 10.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Inc., New York (1982)Google Scholar
  11. 11.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)Google Scholar
  12. 12.
    Birgin, E.G., Castillo, R.A., Martínez, J.M.: Numerical comparison of augmented Lagrangian algorithms for non-convex problems. Comput. Optim. Appl. 31(1), 31–55 (2005)CrossRefGoogle Scholar
  13. 13.
    Birgin, E.G., Fernández, D., Martínez, J.M.: The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems. Optim. Method Softw. 27(6), 1001–1024 (2012)CrossRefGoogle Scholar
  14. 14.
    Dussault, J.P.: Augmented non-quadratic penalty algorithms. Math. Program. 99, 467–486 (2004)CrossRefGoogle Scholar
  15. 15.
    Echebest, N., Sánchez, M.D., Schuverdt, M.L.: Convergence results of an augmented Lagrangian method using the exponential penalty function. J. Optim. Theory. Appl 168, 92–108 (2016)CrossRefGoogle Scholar
  16. 16.
    Haeser, G., Schuverdt, M.L.: On approximate KKT condition and its extension to continuous variational inequalities. J. Optim. Theory. Appl 23, 707–716 (2011)Google Scholar
  17. 17.
    Hestenes, M.R.: Optimization Theory: The Finite Dimensional Case. John Wiley, New York (1975)Google Scholar
  18. 18.
    Kočvara, M., Stingl, M.: PENNON: a code for convex nonlinear and semidefinite programming. Optim. Methods Softw. 18(3), 317–333 (2003)CrossRefGoogle Scholar
  19. 19.
    Martínez, J.M., Birgin, E.G.: Practical augmented Lagrangian methods for constrained optimization. SIAM Publications, Philadelphia (2014)Google Scholar
  20. 20.
    Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429–440 (2011)CrossRefGoogle Scholar
  21. 21.
    Mosheyer, L., Zibulevsky, M.: Penalty/barrier multiplier algorithm for semidefinite programming. Optim. Methods Softw. 13(4), 235–261 (2000)CrossRefGoogle Scholar
  22. 22.
    Tseng, P., Bertsekas, D.P.: On the convergence of the exponential multiplier method for convex programming. Math. Program. 60, 1–19 (1993)CrossRefGoogle Scholar

Copyright information

© Operational Research Society of India 2019

Authors and Affiliations

  1. 1.CONICET, Department of Mathematics, FCEUniversity of La PlataLa PlataArgentina

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