Computational Optimization and Applications

, Volume 65, Issue 3, pp 699–721 | Cite as

Sequential equality-constrained optimization for nonlinear programming

  • E. G. Birgin
  • L. F. Bueno
  • J. M. Martínez


A novel idea is proposed for solving optimization problems with equality constraints and bounds on the variables. In the spirit of sequential quadratic programming and sequential linearly-constrained programming, the new proposed approach approximately solves, at each iteration, an equality-constrained optimization problem. The bound constraints are handled in outer iterations by means of an augmented Lagrangian scheme. Global convergence of the method follows from well-established nonlinear programming theories. Numerical experiments are presented.


Nonlinear programming Sequential equality-constrained optimization Augmented Lagrangian Numerical experiments 



This work was supported by PRONEX-CNPq/FAPERJ E-26/111.449/2010-APQ1, FAPESP (Grants 2010/10133-0, 2013/03447-6, 2013/05475-7, 2013/07375-0, and 2015/02528-8), and CNPq (Grants 309517/2014-1 and 303750/2014-6).


  1. 1.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60, 627–641 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math, Program. 135, 255–273 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new weak constraint qualifications and applications. SIAM J. Optim. 22, 1109–1135 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26, 96–110 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Andreani, R., Martínez, J.M., Schuverdt, M.L.: On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Banihashemi, N., Kaya, C.Y.: Inexact restoration for Euler discretization of box-constrained optimal control problems. J. Optim. Theory Appl. 156, 726–760 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Birgin, E.G., Martínez, J.M.: Improving ultimate convergence of an augmented Lagrangian method. Optim. Methods Softw. 23, 177–195 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM, Philadelphia (2014)CrossRefMATHGoogle Scholar
  10. 10.
    Birgin, E.G., Castelani, E.V., Martinez, A.L.M., Martínez, J.M.: Outer trust-region method for constrained optimization. J. Optim. Theory Appl. 150, 142–155 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bueno, L.F., Haeser, G., Martínez, J.M.: A flexible inexact restoration method for constrained optimization. J. Optim. Theory Appl. 165, 188–208 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Castelani, E.V., Martinez, A.L., Martínez, J.M., Svaiter, B.F.: Addressing the greediness phenomenon in nonlinear programming by means of proximal augmented Lagrangians. Comput. Optim. Appl. 46, 229–245 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Lancelot: A Fortran Package for Large Scale Nonlinear Optimization. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  14. 14.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fischer, A., Friedlander, A.: A new line search inexact restoration approach for nonlinear programming. Comput. Optim. Appl. 46, 333–346 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (2000)CrossRefMATHGoogle Scholar
  17. 17.
    Francisco, J.B., Martínez, J.M., Martínez, L., Pisnitchenko, F.: Inexact Restoration method for minimization problems arising in electronic structure calculations. Comput. Optim. Appl. 50, 555–590 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkings University Press, Baltimore (1996)MATHGoogle Scholar
  19. 19.
    Gonzaga, C.C., Karas, E.W., Vanti, M.: A globally convergent filter method for nonlinear programming. SIAM J. Optim. 14, 646–669 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gould, N.I.M., Orban, D., Toint, PhL: CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput. Optim. Appl. 60, 545–557 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    HSL: A collection of Fortran codes for large scale scientific computation. (2013)
  22. 22.
    Karas, E.W., Gonzaga, C.C., Ribeiro, A.A.: Local convergence of filter methods for equality constrained non-linear programming. Optimization 59, 1153–1171 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Karas, E.W., Pilotta, E.A., Ribeiro, A.A.: Numerical comparison of merit function with filter criterion in inexact restoration algorithms using hard-spheres problems. Comput. Optim. Appl. 44, 427–441 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kaya, C.Y.: Inexact restoration for Runge-Kutta discretization of optimal control problems. SIAM J. Numer. Anal. 48, 1492–1517 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kaya, C.Y., Martínez, J.M.: Euler discretization and inexact restoration for optimal control. J. Optim. Theory Appl. 134, 191–206 (2007)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mangasarian, O.L., Fromovitz, S.: The Fritz-John necessary optimality conditions in presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Martínez, J.M.: Inexact-restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming. J. Optim. Theory Appl. 111, 39–58 (2001)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Martínez, J.M., Pilotta, E.A.: Inexact-restoration algorithms for constrained optimization. J. Optim. Theory Appl. 104, 135–163 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Murtagh, B.A., Saunders, M.A.: Large-scale linearly constrained optimization. Math. Program. 14, 41–72 (1978)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  31. 31.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Qi, L., Wei, Z.: On the constant linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer Science, Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil
  2. 2.Institute of Science and TechnologyFederal University of São PauloSão José dos CamposBrazil
  3. 3.Department of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific ComputingState University of CampinasCampinasBrazil

Personalised recommendations