Skip to main content
SpringerLink
Log in

On the Global Convergence of a Projective Trust Region Algorithm for Nonlinear Equality Constrained Optimization

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A trust-region sequential quadratic programming (SQP) method is developed and analyzed for the solution of smooth equality constrained optimization problems. The trust-region SQP algorithm is based on filter line search technique and a composite-step approach, which decomposes the overall step as sum of a vertical step and a horizontal step. The algorithm includes critical modifications of horizontal step computation. One orthogonal projective matrix of the Jacobian of constraint functions is employed in trust-region subproblems. The orthogonal projection gives the null space of the transposition of the Jacobian of the constraint function. Theoretical analysis shows that the new algorithm retains the global convergence to the first-order critical points under rather general conditions. The preliminary numerical results are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahookhosh, M., Amini, K., Peyghami, M. R.: A nonmonotone trust-region line search method for largescale unconstrained optimization. Appl. Math. Model., 36, 478–487 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, Y., Sun, W.: A dwindling filter line search method for unconstrained optimization. Math. Comp., 84, 187–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Z., Qiu, S., Jiao, Y.: A penalty-free method for equality constrained optimization. J. Ind. Manag. Optim., 9, 391–409 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Conn, A. R., Gould, N. I. M., Toint, P. L.: Trust-region methods, MPS-SIAM Series on Optimization 1, SIAM, Philadelphia, 2000

    Google Scholar 

  5. Cui, Z., Wu, B., Qu, S.: Combining nonmonotone conic trust region and line search techniques for unconstrained optimization. J. Comput. Appl. Math., 235, 2432–2441 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dolan, E. D., Moré, J. J.: Benchmarking optimization software with performance profiles. Math. Program., 91, 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fletcher, R., Gould, N. I. M., Leyffer, S., et al.: Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming. SIAM J. Optim., 13, 635–659 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program., 91, 239–269 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fletcher, R., Leyffer, S., Toint, P. L.: On the global convergence of a filter-SQP algorithm. SIAM J. Optim., 13, 44–59 (electronic) (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gill, P. E., Murray, W., Saunders, M. A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev., 47, 99–131 (electronic) (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gould, N. I. M., Toint, P. L.: Nonlinear programming without a penalty function or a filter. Math. Program., 122, 155–196 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gu, C., Zhu, D.: A dwindling filter line search algorithm for nonlinear equality constrained optimization. J. Syst. Sci. Complex., 28, 623–637 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gu, C., Zhu, D.: A nonmonotone line search filter method with reduced Hessian updating for nonlinear optimization. J. Syst. Sci. Complex., 26, 534–555 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gu, C., Zhu, D.: A secant algorithm with line search filter method for nonlinear optimization. Appl. Math. Model., 35, 879–894 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gu, C., Zhu, D. T.: Global and local convergence of a new affine scaling trust region algorithm for linearly constrained optimization. Acta Math. Sin., Engl. Ser., 32, 1203–1213 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heinkenschloss, M., Ridzal, D.: A matrix-free trust-region SQP method for equality constrained optimization. SIAM J. Optim., 24, 1507–1541 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hock, W., Schittkowski, K.: Test examples for nonlinear programming codes, volume 187 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin-New York, 1981

    Book  MATH  Google Scholar 

  18. Lalee, M., Nocedal, J., Plantenga, T.: On the implementation of an algorithm for large-scale equality constrained optimization. SIAM J. Optim., 8, 682–706 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, X., Yuan, Y.: A sequential quadratic programming method without a penalty function or a filter for nonlinear equality constrained optimization. SIAM J. Optim., 21, 545–571 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nocedal, J., Wright, S. J.: Numerical Optimization (second edition), Springer, New York, 2006

    MATH  Google Scholar 

  21. Nocedal, J., Yuan, Y. X.: Combining trust region and line search techniques. In Advances in nonlinear programming (Beijing, 1996), volume 14 of Appl. Optim., 153–175. Kluwer Acad. Publ., Dordrecht (1998)

    Chapter  Google Scholar 

  22. Pei, Y., Zhu, D.: A trust-region algorithm combining line search filter method with Lagrange merit function for nonlinear constrained optimization. Appl. Math. Comput., 247, 281–300 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Pei, Y., Zhu, D.: A trust-region algorithm combining line search filter technique for nonlinear constrained optimization. Int. J. Comput. Math., 91, 1817–1839 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Powell, M. J. D., Yuan, Y.: A trust region algorithm for equality constrained optimization. Math. Programming, 49, 189–211 (1990/91)

    Article  MathSciNet  MATH  Google Scholar 

  25. Qiu, S., Chen, Z.: Global and local convergence of a class of penalty-free-type methods for nonlinear programming. Appl. Math. Model., 36, 3201–3216 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Qu, S. J., Zhang, Q. P., Yang, Y. T.: A nonmonotone conic trust region method based on line search for solving unconstrained optimization. J. Comput. Appl. Math., 224, 514–526 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schittkowski, K.: More test examples for nonlinear programming codes, volume 282 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1987

    Book  Google Scholar 

  28. Shen, C., Leyffer, S., Fletcher, R.: A nonmonotone filter method for nonlinear optimization. Comput. Optim. Appl., 52, 583–607 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Su, K., An, H.: Global convergence of a nonmonotone filter method for equality constrained optimization. Appl. Math. Comput., 218, 9396–9404 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Sun, W., Yuan, Y. X.: Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006

    MATH  Google Scholar 

  31. Ulbrich, M., Ulbrich, S., Vicente, L. N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Math. Program., 100, 379–410 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wächter, A., Biegler, L. T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim., 16, 1–31 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, H., Pu, D.: A nonmonotone filter trust region method for the system of nonlinear equations. Appl. Math. Model., 37, 498–506 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yuan, G., Wei, Z., Zhang, M.: An active-set projected trust region algorithm for box constrained optimization problems. J. Syst. Sci. Complex., 28, 1128–1147 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, J., Zhu, D.: A trust region typed dogleg method for nonlinear optimization. Optimization, 21, 543–557 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhou, Q., Hang, D.: Nonmonotone adaptive trust region method with line search based on new diagonal updating. Appl. Numer. Math., 91, 75–88 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhu, D.: An affine scaling trust-region algorithm with interior backtracking technique for solving boundconstrained nonlinear systems. J. Comput. Appl. Math., 184, 343–361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhu, D.: Superlinearly convergent affine scaling interior trust-region method for linear constrained lc 1 minimization. Acta Math. Sin., Engl. Ser., 24, 2081–2100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu, X., Pu, D.: A restoration-free filter SQP algorithm for equality constrained optimization. Appl. Math. Comput., 219, 6016–6029 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are very grateful to the reviewers for their valuable and insightful comments and suggestions, which have aided us in improving the presentation of this paper.

Author information

Authors and Affiliations

  1. Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, College of Mathematics and Information Science, He’nan Normal University, Xinxiang, 453007, P. R. China

    Yong Gang Pei

  2. Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, P. R. China

    De Tong Zhu

Authors
  1. Yong Gang Pei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. De Tong Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Gang Pei.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 11671122 and 11371253), Key Scientific Research Project for Colleges and Universities in He’nan Province (Grant No. 15A110031), Key Scientific and Technological Project of He’nan Province (Grant No. 162102210069), Natural Science Foundation of He’nan Normal University (Grant No. 2014QK04) and Ph. D. Research Foundation of He’nan Normal University (Grant Nos. QD13041 and QD14155)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pei, Y.G., Zhu, D.T. On the Global Convergence of a Projective Trust Region Algorithm for Nonlinear Equality Constrained Optimization. Acta. Math. Sin.-English Ser. 34, 1804–1828 (2018). https://doi.org/10.1007/s10114-018-7063-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-018-7063-4

Keywords

MR(2010) Subject Classification

Search

Navigation