Skip to main content
SpringerLink
Log in

A new ɛ-generalized projection method of strongly sub-feasible directions for inequality constrained optimization

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

In this paper, the nonlinear optimization problems with inequality constraints are discussed. Combining the ideas of the strongly sub-feasible directions method and the ɛ-generalized projection technique, a new algorithm starting with an arbitrary initial iteration point for the discussed problems is presented. At each iteration, the search direction is generated by a new ɛ-generalized projection explicit formula, and the step length is yielded by a new Armijo line search. Under some necessary assumptions, not only the algorithm possesses global and strong convergence, but also the iterative points always get into the feasible set after finite iterations. Finally, some 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. G. Zoutendijk, Methods of Feasible Directions, Elsevier, Amsterdam, 1960.

    MATH  Google Scholar 

  2. J. B. Rosen, The gradient projection method for nonlinear programming, part I, linear constraints, SIAM J. Appl. Math., 1960, 8(1): 181–217.

    Article  MATH  Google Scholar 

  3. D. M. Topkis and A. F. Veinott, On the convergence of some feasible direction algorithms for nonlinear programming, SIAM J. Control, 1967, 5(2): 268–279.

    Article  MathSciNet  MATH  Google Scholar 

  4. O. Pironneau and E. Polak, Rate of convergence of a class of method of feasible directions, SIAM J. Num. Anal., 1973, 1(10): 161–173.

    Article  MathSciNet  Google Scholar 

  5. D. Z. Du and J. Sun, A new gradient projection method, Math. Numer. Sinica, 1983, 4: 378–386.

    MathSciNet  Google Scholar 

  6. Z. Y. Gao and G. P. He, A generalized gradient projection method for constrained optimization, Kexue Tongbao, 1991, 19: 1444–1447.

    Google Scholar 

  7. Y. L. Lai and J. B. Jian, A generalized gradient projection method with initial point for the optimized problem with nonlinear constraints, J. Sys. Sci. and Math. Scis., 1995, 15(4): 374–380.

    MathSciNet  MATH  Google Scholar 

  8. D. Z. Du, A gradient projection algorithm for nonlinear constraints, J. Acta Math. Appl. Sini., 1985, 8: 7–16.

    MATH  Google Scholar 

  9. S. J. Xue and J. B. Jian, A generalized gradient projection method under nonlinear constraints, J. Jinan Univ., 1997, 18(1): 27–32.

    Google Scholar 

  10. D. Z. Du, A modification of Rosen-Polak’s algorithm, Kexue Tongbao, 1983, 28: 301–305.

    MATH  Google Scholar 

  11. E. Polak, R. Trhan, and D. Q. Mayne, Combined Phase I-Phase II methods of feasible directions, Math. Prog., 1979, 17: 61–73.

    Article  MATH  Google Scholar 

  12. J. B. Jian, Strong combined Phase I-Phase II methods of sub-feasible directions, J. Math. Econom., 1995, 12(1): 64–70.

    MathSciNet  Google Scholar 

  13. J. B. Jian, Researches on superlinearly and quadratically convergent algorithms for nonlinearly constrained optimization, Ph. D. Thesis, School of Xi’an Jiaotong Univ., Xi’an, China, 2000.

    Google Scholar 

  14. J. B. Jian and K. C. Zhang, Sub-feasible direction method with strong convergence for inequality constrained optimization, J. Xi’ an Jiaotong Univ., 1999, 33(8): 88–103.

    MathSciNet  MATH  Google Scholar 

  15. J. B. Jian and Y. M. Liang, Finitely convergent algorithm of generalized gradient projection for systems of nonlinear inequalities, Neural, Parallel and Sci. Comput., 2004, 12: 207–218.

    MathSciNet  MATH  Google Scholar 

  16. M. Fukushima, A finitely convergent algorithm for convex inequalities, IEEE Trams. Autom. Contr., 1982, 27(5): 1126–1127.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. B. Jian, X. L. Zhang, and R. Quan, A new finitely convergent algorithm for systems of nonlinear inequalities, Appl. Math. Letter, 2007, 20(4): 405–411.

    Article  MathSciNet  MATH  Google Scholar 

  18. O. A. Elwakeil and J. S. Arora, Methods for finding feasible points in constrained optimization, AIAA Journal, 1995, 33(9): 1715–1719.

    Article  MATH  Google Scholar 

  19. J. B. Jian, W. X. Cheng, and X. Y. Ke, Finitely convergent ɛ-generalized projection algorithm for nonlinear systems, J. Math. Anal. Appl., 2007, 332(2): 1445–1458.

    Article  MathSciNet  Google Scholar 

  20. W. Hock and K. Schittkowski, Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems, Springer, 1981.

  21. I. Bongartz, A. R. Conn, N. I. Gould, and PH. L. Toint, CUTE: Constrained and unconstrained testing environment, ACM Tras. Math. Software, 1995, 21: 123–160.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Jinbao Jian

  2. Department of Mathematics and Computer Science, Yulin Normal University, Yulin, 537000, China

    Guodong Ma

  3. College of Science, Shanghai University, Shanghai, 200444, China

    Chuanhao Guo

Authors
  1. Jinbao Jian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guodong Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chuanhao Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinbao Jian.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant Nos. 71061002 and 10771040, the Project supported by Guangxi Science Foundation under Grant No. 0832052, and Science Foundation of Guangxi Education Department under Grant No. 200911MS202.

This paper was recommended for publication by Editor Xiaoguang YANG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jian, J., Ma, G. & Guo, C. A new ɛ-generalized projection method of strongly sub-feasible directions for inequality constrained optimization. J Syst Sci Complex 24, 604–618 (2011). https://doi.org/10.1007/s11424-011-8105-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-011-8105-5

Key words

Search

Navigation