Skip to main content
SpringerLink
Log in

A new criterion for the inexact logarithmic-quadratic proximal method and its derived hybrid methods

  • Original Paper
  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

To solve nonlinear complementarity problems, the inexact logarithmic-quadratic proximal (LQP) method solves a system of nonlinear equations (LQP system) approximately at each iteration. Therefore, the efficiencies of inexact-type LQP methods depend greatly on the involved inexact criteria used to solve the LQP systems. This paper relaxes inexact criteria of existing inexact-type LQP methods and thus makes it easier to solve the LQP system approximately. Based on the approximate solutions of the LQP systems, a descent method, and a prediction–correction method are presented. Convergence of the new methods are proved under mild assumptions. Numerical experiments for solving traffic equilibrium problems demonstrate that the new methods are more efficient than some existing methods and thus verify that the new inexact criterion is attractive in practice.

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. Auslender A. and Haddou M. (1995). An interior proximal point method for convex linearly constrained problems and its extension to variational inequalities. Math. Program. 71: 77–100

    Google Scholar 

  2. Auslender A., Teboulle M. and Ben-Tiba S. (1999). A logarithmic-quadratic proximal method for variational inequalities. Comput. Optim. Appl. 12: 31–40

    Article  Google Scholar 

  3. Bauschke H.H. and Combettes P.L. (2001). A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26(2): 248–264

    Article  Google Scholar 

  4. Burachik R.S. and Iusem A.N. (1998). A generalized proximal point alogrithm for the variational inequality problem in a Hilbert space. SIAM J. Optim. 8: 197–216

    Google Scholar 

  5. Burachik R.S. and Svaiter B.F. (2001). A relative error tolerance for a family of generalized proximal point methods. Math. Oper. Res. 26(4): 816–831

    Article  Google Scholar 

  6. Eaves B.C. (1971). On the basic theorem of complementarity. Math. Program. 1: 68–75

    Article  Google Scholar 

  7. Eckstein J. (1998). Approximating iterations in Bregman-function-based proximal algorithms. Math. Program. 83: 113–123

    Google Scholar 

  8. Güler O. (1991). On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29: 403–419

    Article  Google Scholar 

  9. Han D.R. and He B.S. (2001). A new accuarcy criterion for approximate proximal point algorithms. J. Math. Anal. Appl. 263: 343–353

    Article  Google Scholar 

  10. Harker P.T. and Pang J.S. (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program. 48: 161–220

    Article  Google Scholar 

  11. He B.S., Liao L.Z. and Yang Z.H. (2003). A new approximate proximal point algorithm for maximal monotone operator. Sci. China Ser. A 46(2): 200–206

    Article  Google Scholar 

  12. He B.S., Yang Z.H. and Yuan X.M. (2004). An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300(2): 362–374

    Article  Google Scholar 

  13. He B.S., Liao L.Z. and Yuan X.M. (2006). A LQP method-based interior prediction-correction for nonliner complementarity problems. J. Comput. Math. 24(1): 33–44

    Google Scholar 

  14. He B.S., Xu Y. and Yuan X.M. (2006). A Logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities. Comput. Optim. Appl. 35: 19–46

    Article  Google Scholar 

  15. Martinet B. (1970). Regularization d’inéquations variationnelles par approximations succesives. Rev. Fr. Inform. Rech. Opérat. 2: 154–159

    Google Scholar 

  16. Moré J.J. (1996). Global methods for nonlinear complementarity problems. Math. Oper. Res. 21: 589–614

    Article  Google Scholar 

  17. Rockafellar R.T. (1976). Monotone operators and the proximal point algorithm. SIAM J. Optim. 126: 877–898

    Google Scholar 

  18. Teboulle M. (1997). Convergence of proximal-like algorithms. SIAM J. Optim. 7: 1069–1083

    Article  Google Scholar 

  19. Xu, Y., Bnouhachem, A.: A new logarithmic-quadratic proximal prediction–correction method for NCP. manuscript (2006)

  20. Yang H. and Bell M.G.H. (1997). Traffic restraint, road pricing and network equilibrium. Transport. Res. B 31: 303–314

    Article  Google Scholar 

  21. Yuan X.M. (2007). The prediction–correction approach to nonlinear complementarity problems. Eur. J. Oper. Res. 176(3): 1357–1370

    Article  Google Scholar 

  22. Zarantonello E.H. (1971). Projection on covex sets in Hilbert space and sepctral theory. In: Zarantonello, E.H. (eds) Contributation to Nonlinear Functional Analysis, pp 00–00. Academic Press, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Management Science, Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, 200052, China

    Xiao-Ming Yuan

Authors
  1. Xiao-Ming Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Ming Yuan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yuan, XM. A new criterion for the inexact logarithmic-quadratic proximal method and its derived hybrid methods. J Glob Optim 40, 529–543 (2008). https://doi.org/10.1007/s10898-006-9123-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-006-9123-z

Keywords

Search

Navigation