Computational Optimization and Applications

, Volume 53, Issue 2, pp 425–452 | Cite as

Active-set Newton methods for mathematical programs with vanishing constraints

  • A. F. Izmailov
  • A. L. Pogosyan


Mathematical programs with vanishing constraints constitute a new class of difficult optimization problems with important applications in optimal topology design of mechanical structures. Vanishing constraints usually violate standard constraint qualifications, which gives rise to serious difficulties in theoretical and numerical treatment of these problems. In this work, we suggest several globalization strategies for the active-set Newton-type methods developed earlier by the authors for this problem class, preserving superlinear convergence rate of these methods under weak assumptions. Preliminary numerical results demonstrate that our approach is rather promising and competitive with respect to the existing alternatives.


Mathematical program with vanishing constraints Constraint qualification Optimality condition Active-set method Sequential quadratic programming Semismooth Newton method Global convergence 



The authors thank the anonymous referees for useful comments. In particular, Remark 4.1 is in response to the question raised by one of the referees.


  1. 1.
    Achtziger, W., Kanzow, C.: Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math. Program. 114, 69–99 (2007) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Achtziger, W., Hoheisel, T., Kanzow, C.: A smoothing-regularization approach to mathematical programs with vanishing constraints. Preprint 284, Institute of Mathematics, University of Würzburg (2008) Google Scholar
  3. 3.
    Bonnans, J.F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chen, Y.D., Gao, Y., Liu, Y.-J.: An inexact SQP Newton method for convex SC1 minimization problems. J. Optim. Theory Appl. 146, 33–49 (2010) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) MATHGoogle Scholar
  6. 6.
    De Luca, T., Facchinei, F., Kanzow, C.: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16, 173–205 (2000) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  9. 9.
    Facchinei, F., Fischer, A., Kanzow, C.: Regularity properties of a semismooth reformulation of variational inequalities. SIAM J. Optim. 8, 850–869 (1998) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J. Optim. 9, 14–32 (1999) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fischer, A.: Local behaviour of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94, 91–124 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hager, W.W., Gowda, M.S.: Stability in the presence of degeneracy and error estimation. Math. Program. 85, 181–192 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hoheisel, T., Kanzow, C.: First- and second-order optimality conditions for mathematical programs with vanishing constraints. Appl. Math. 52, 495–514 (2007) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hoheisel, T., Kanzow, C.: Stationarity conditions for mathematical programs with vanishing constraints using weak constraint qualifications. J. Math. Anal. Appl. 337, 292–310 (2008) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hoheisel, T., Kanzow, C.: On the Abadie and Guignard constraint qualifications for mathematical programs with vanishing constraints. Optimization 58, 431–448 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hoheisel, T., Kanzow, C., Outrata, J.: Exact penalty results for mathematical programs with vanishing constraints. Nonlinear Anal. 72, 2514–2526 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hoheisel, T., Kanzow, C., Schwartz, A.: Convergence of a local regularization approach for mathematical programs with complementarity or vanishing constraints. Optim. Methods Softw. doi: 10.1080/10556788.2010.535170
  18. 18.
    Izmailov, A.F., Pogosyan, A.L.: Optimality conditions and Newton-type methods for mathematical programs with vanishing constraints. Comput. Math. Math. Phys. 49, 1128–1140 (2009) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Izmailov, A.F., Pogosyan, A.L.: On active-set methods for mathematical programs with vanishing constraints. In: Bereznyov, V.A. (ed.) Theoretical and Applied Problems of Nonlinear Analysis, pp. 18–49. CCAS, Moscow (2009) (In Russian) Google Scholar
  20. 20.
    Izmailov, A.F., Solodov, M.V.: An active-set Newton method for mathematical programs with complementarity constraints. SIAM J. Optim. 19, 1003–1027 (2008) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Izmailov, A.F., Solodov, M.V.: Mathematical programs with vanishing constraints: optimality conditions, sensitivity, and a relaxation method. J. Optim. Theory Appl. 142, 501–532 (2009) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Izmailov, A.F., Pogosyan, A.L., Solodov, M.V.: Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints. Comput. Optim. Appl. 51, 199–221 (2012) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge Univ. Press, Cambridge (1996) CrossRefGoogle Scholar
  24. 24.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006) MATHGoogle Scholar
  25. 25.
    Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic, Boston (1998) MATHGoogle Scholar
  26. 26.
    Tseng, P.: Growth behavior of a class of merit functions for the nonlinear complementarity problem. J. Optim. Theory Appl. 89, 17–37 (1996) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.VMK Faculty, OR DepartmentMoscow State University, MSUMoscowRussia

Personalised recommendations