An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities

  • Paul ArmandEmail author
  • Ngoc Nguyen Tran


We present a primal-dual augmented Lagrangian method for solving an equality constrained minimization problem, which is able to rapidly detect infeasibility. The method is based on a modification of the algorithm proposed in Armand and Omheni (Optim Methods Softw 32(1):1–21, 2017). A new parameter is introduced to scale the objective function and, in case of infeasibility, to force the convergence of the iterates to an infeasible stationary point. It is shown, under mild assumptions, that whenever the algorithm converges to an infeasible stationary point, the rate of convergence is quadratic. This is a new convergence result for the class of augmented Lagrangian methods. The global convergence of the algorithm is also analyzed. It is also proved that, when the algorithm converges to a stationary point, the properties of the original algorithm are preserved. The numerical experiments show that our new approach is as good as the original one when the algorithm converges to a local minimum, but much more efficient in case of infeasibility.


Nonlinear optimization Augmented Lagrangian method Infeasibility detection 


  1. 1.
    Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Optimization (Symposium of University Keele, Keele, 1968), pp. 283–298. Academic Press, London (1969)Google Scholar
  3. 3.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: LANCELOT, Springer Series in Computational Mathematics, vol. 17. Springer, Berlin (1992). A Fortran package for large-scale nonlinear optimization (release A)Google Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111(1–2), 5–32 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Birgin, E.G., Martínez, J.M.: Practical augmented Lagrangian methods for constrained optimization, Fundamentals of Algorithms, vol. 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2014)Google Scholar
  7. 7.
    Armand, P., Omheni, R.: A globally and quadratically convergent primal-dual augmented Lagrangian algorithm for equality constrained optimization. Optim. Methods Softw. 32(1), 1–21 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Martínez, J.M., Prudente, L.D.F.: Handling infeasibility in a large-scale nonlinear optimization algorithm. Numer. Algorithms 60(2), 263–277 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Birgin, E.G., Martínez, J.M., Prudente, L.F.: Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming. J. Glob. Optim. 58(2), 207–242 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Birgin, E.G., Martínez, J.M., Prudente, L.F.: Optimality properties of an augmented Lagrangian method on infeasible problems. Comput. Optim. Appl. 60(3), 609–631 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Birgin, E.G., Floudas, C.A., Martínez, J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Program. 125(1), 139–162 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gonçalves, M.L.N., Melo, J.G., Prudente, L.F.: Augmented Lagrangian methods for nonlinear programming with possible infeasibility. J. Glob. Optim. 63(2), 297–318 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Byrd, R.H., Curtis, F.E., Nocedal, J.: Infeasibility detection and SQP methods for nonlinear optimization. SIAM J. Optim. 20(5), 2281–2299 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Burke, J.V., Curtis, F.E., Wang, H.: A sequential quadratic optimization algorithm with rapid infeasibility detection. SIAM J. Optim. 24(2), 839–872 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bertsekas, D.P.: Constrained optimization and Lagrange multiplier methods. In: Computer Science and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1982)Google Scholar
  16. 16.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)Google Scholar
  17. 17.
    Forsgren, A., Gill, P.E., Wright, M.H.: Interior methods for nonlinear optimization. SIAM Rev. 44(4), 525–597 (2002). (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Armand, P., Benoist, J., Orban, D.: From global to local convergence of interior methods for nonlinear optimization. Optim. Methods Softw. 28(5), 1051–1080 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Forsgren, A., Gill, P.E.: Primal-dual interior methods for nonconvex nonlinear programming. SIAM J. Optim. 8(4), 1132–1152 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dennis Jr., J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996)CrossRefGoogle Scholar
  21. 21.
    Gould, N.I.M., Orban, D., Toint, P.L.: Cuter and sifdec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Armand, P., Benoist, J., Omheni, R., Pateloup, V.: Study of a primal-dual algorithm for equality constrained minimization. Comput. Optim. Appl. 59(3), 405–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Duff, I.S.: MA57–a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw. 30(2), 118–144 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Birgin, E.G., Bueno, L.F., Martínez, J.M.: Sequential equality-constrained optimization for nonlinear programming. Comput. Optim. Appl. 65(3), 699–721 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Curtis, F.E.: A penalty-interior-point algorithm for nonlinear constrained optimization. Math. Program. Comput. 4(2), 181–209 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nocedal, J., Öztoprak, F., Waltz, R.A.: An interior point method for nonlinear programming with infeasibility detection capabilities. Optim. Methods Softw. 29(4), 837–854 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Armand, P., Omheni, R.: A mixed logarithmic barrier-augmented Lagrangian method for nonlinear optimization. J. Optim. Theory Appl. 173(2), 523–547 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Armand, P., Tran, N.N.: Rapid infeasibility detection in a mixed logarithmic barrier augmented Lagrangian method for nonlinear optimization. Optim. Methods Softw. (2018).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Limoges - Laboratoire XLIMLimogesFrance

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