Convergence of a stabilized SQP method for equality constrained optimization

  • Songqiang QiuEmail author


We herein present a stabilized sequential programming method for equality constrained programming. The proposed method uses the concepts of proximal point methods and primal-dual regularization. A sequence of regularized problems are approximately solved with the regularization parameter approaching zero. At each iteration, a regularized QP subproblem is solved to obtain a primal-dual search direction. Further, a trust-funnel-like line search scheme is used to globalize the algorithm, and a global convergence under the weak assumption of cone-continuity property is shown. To achieve a fast local convergence, a specially designed second-order correction (SOC) technique is adopted near a solution. Under the second-order sufficient condition and some weak conditions (among which no constraint qualification is involved), the regularized QP subproblem transits to a stabilized QP subproblem in the limit. By possibly combining with the SOC step, the full step will be accepted in the limit and hence the superlinearly local convergence is achieved. Preliminary numerical results are reported, which are encouraging.


Equality constrained optimization Stabilized sequential quadratic programming Trust-funnel-like method Global convergence Local convergence 

Mathematics Subject Classification

90C30 90C55 65K05 



The authors would like to thank the two anonymous referees for their useful comments that helped to improve the original version of the manuscript.


  1. 1.
    Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60, 627–641 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and alalgorithm consequences. SIAM J. Optim. 26(1), 96–110 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. Fundamental Algorithm. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
  4. 4.
    Boggs, P.T., Tolle, J.W.: Sequential quadratic programming. Acta Numer. 4, 1–51 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Curtis, F.E., Gould, N.I.M., Robinson, D.P., Toint, PhL: An interior-point trust-funnel algorithm for nonlinear optimization. Math. Program. 161(1), 73–134 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
  7. 7.
    Fernández, D., Pilotta, E.A., Torres, G.A.: An inexact restoration strategy for the globalization of the sSQP method. Comput. Optim. Appl. 54(3), 595–617 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fernández, D., Solodov, M.V.: Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variatinal problems. Math. Program. 125(1, Ser. A), 47–73 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program. 91(2), 239–269 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. The Scientific Press, Singapore (1993)zbMATHGoogle Scholar
  11. 11.
    Friedlander, M.P., Orban, D.: A primal-dual regularized interior-point method for convex quadratic programs. Math. Program. Comput. 4(1), 71–107 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gill, P.E., Kungurtsev, V., Robinson, D.P.: A stabilized SQP mehod: global convergence. IMA J. Numer. Anal. 37, 407–443 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gill, P.E., Kungurtsev, V., Robinson, D.P.: A stabilized SQP method: superlinear convergence. Math. Program. 163(1), 369–410 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gill, P.E., Robinson, D.P.: A globally convergent stabilized SQP method. SIAM J. Optim. 23(4), 1983–2010 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gill, P.E., Wong, E.: Sequential quadratic programming methods. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 147–224. Springer, New York, NY (2012)Google Scholar
  17. 17.
    Gondzio, J.: Matrix-free interior point method. Comput. Optim. Appl. 51(2), 457–480 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gould, N.I.M., Toint, PhL: Nonlinear programming without a penalty function or a filter. Math. Program. 122(1), 155–196 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hager, W.W.: Stabilized sequential quadratic programming. Comput. Optim. Appl. 12(1–3), 253–273 (1999)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Code. Lecture Notes in Economics and Mathematical Systems, 187. Springer, Berlin (1987)zbMATHGoogle Scholar
  21. 21.
    Izmailov, A.F., Krylova, A., Uskov, E.I.: Hybrid globalization of stabilized sequential quadratic programming method. In: Bereznyov, V. (ed.) Theoretical and Applied Problems of Nonlinear Analysis, pp. 47–66. Computing Center RAS, Moscow (2011). (in Russian)Google Scholar
  22. 22.
    Izmailov, A.F., Solodov, M.V.: Newton-type methods for optimization problems without constraint qualifications. SIAM J. Optim. 15(1), 210–228 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Izmailov, A.F., Solodov, M.V., Uskov, E.I.: Globalizing stabilized sequential quadratic programming method by smooth primal-dual exact penalty function. J. Optim. Theory Appl. 169(1), 148–178 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Izmailov, A.F., Solodov, M.V.: Stabilized SQP revisited. Math. Program. 133, 93–120 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Izmailov, A.F., Uskov, E.I.: Subspace-stabilized sequential quadratic programming. Comput. Optim. Appl. 67(1), 129–154 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Maratos, N.: Exact penalty function algorithms for finite dimensional and control optimization problems. Ph.D. thesis, University of London (1978)Google Scholar
  27. 27.
    Martíne, J.M., Pilotta, E.A.: Inexact restoration methods for nonlinear programming: advances and perspectives. In: Qi, L., Teo, K., Yang, X. (eds.) Optimization and Control with Applications, Applied Optimization, vol. 96, pp. 271–291. Springer, New York (2005)Google Scholar
  28. 28.
    Qi, L.Q., Wei, Z.X.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10(4), 963–981 (2000)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Qiu, S.Q., Chen, Z.W.: Global and local convergence of a class of penalty-free-type methods for nonlinear programming. Appl. Math. Model. 36(7), 3201–3216 (2012)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Qiu, S.Q., Chen, Z.W.: A globally convergent penalty-free method for optimization with equality constraints and simple bounds. Acta Appl. Math. 142(1), 39–60 (2016)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Rockafellar, R.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1(2), 97–116 (1976)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Rusten, T., Winther, R.: A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13(3), 887–904 (1992)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Shen, C.G., Zhang, L.H., Liu, W.: A stabilized filter SQP algorithm for nonlinear programming. J. Glob. Optim. 65, 677–708 (2016)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Silvester, D., Wathen, A.: Fast iterative solution of stabilised stokes systems part ii: Using general block preconditioners. SIAM J. Numer. Anal. 31(5), 1352–1367 (1994)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Sun, W.Y., Yuan, Y.X.: Optimization Theory and Methods: Nonlinear Programming. Springer, Berlin (2006)zbMATHGoogle Scholar
  36. 36.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Wright, S.J.: Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. 11(3), 253–275 (1998)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wright, S.J.: Modifying SQP for degenerate problems. SIAM J. Optim. 13(2), 470–497 (2002)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Wright, S.J.: An algorithm for degenerate nonlinear programming with rapid local convergence. SIAM J. Optim. 15(3), 673–696 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouChina

Personalised recommendations