Skip to main content
SpringerLink
Log in

Prediction-correction method with BB step sizes

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

In the prediction-correction method for variational inequality (VI) problems, the step size selection plays an important role for its performance. In this paper, we employ the Barzilai-Borwein (BB) strategy in the prediction step, which is efficient for many optimization problems from a computational point of view. To guarantee the convergence, we adopt the line search technique, and relax the conditions to accept the BB step sizes as large as possible. In the correction step, we utilize a longer step length to calculate the next iteration point. Finally, we present some preliminary numerical results to show the efficiency of the algorithms.

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. Barzilai J, Borwein J W. Two-point step size gradient methods. IMA J Numer Anal, 1988, 8: 141–148

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas D P, Gafni E M. Projection methods for variational inequalities with applications to the traffic assignment problem. Math Program Study, 1982, 17: 139–159

    Article  MathSciNet  MATH  Google Scholar 

  3. Birgin E G, Chambouleyron I, Martinez J M. Estimation of the optical constants and the thickness of thin films using unconstrained optimization. J Comput Phys, 1999, 151: 862–880

    Article  MATH  Google Scholar 

  4. Dafermos S. Traffc equilibrium and variational inequalities. Transp Sci, 1980, 14: 42–54

    Article  Google Scholar 

  5. Dai Y H, Fletcher R. Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer Math, 2005, 100: 21–47

    Article  MathSciNet  MATH  Google Scholar 

  6. Dai Y H, Fletcher R. New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds. Math Program, 2006, 106: 403–421

    Article  MathSciNet  MATH  Google Scholar 

  7. Dai Y H, Hager W W, Schittkowski K, Zhang H C. The cyclic Barzilai-Borwein method for unconstrained optimization. IMA J Numer Anal, 2006, 26: 604–627

    Article  MathSciNet  MATH  Google Scholar 

  8. Dai Y H, Yuan Y X. Alternate minimization gradient method. IMA J Numer Anal, 2003, 23: 377–393

    Article  MathSciNet  MATH  Google Scholar 

  9. Facchinei F, Pang J S. Finite-Dimensional Variational Inequalities and Complemen-tarity Problems, Vol 1. Berlin: Springer-Verlag, 2003

  10. Goldstein A A. Convex programming in Hilbert space. Bull Amer Math Soc, 1964, 70: 709–710

    Article  MathSciNet  MATH  Google Scholar 

  11. Hager W W, Mair B A, Zhang H C. An affine-scaling interior-point CBB method for box-constrained optimization. Math Program, 2009, 119: 1–32

    Article  MathSciNet  MATH  Google Scholar 

  12. Hager W W, Zhang H C. A new active set algorithm for box constrained optimization. SIAM J Optim, 2006, 17: 526–557

    Article  MathSciNet  MATH  Google Scholar 

  13. Harker P T, Pang J S. A damped-Newton method for the linear complementarity problem. Lect Appl Math, 1990, 26: 265–284

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. He B S, Liao L Z. Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl, 2002, 112: 111–128

    Article  MathSciNet  MATH  Google Scholar 

  16. He H J, Han D R, Li Z B. Some projection methods with the BB step sizes for variational inequalities. J Comput Appl Math, 2012, 236: 2590–2604

    Article  MathSciNet  MATH  Google Scholar 

  17. Korpelevich G M. An extragradient method for finding saddle points and other problems. Èkon Mat Metody, 1976, 12: 747–756

    MathSciNet  MATH  Google Scholar 

  18. Levitin E S, Polyak B T. Constrained minimization methods. USSR Comput Math Math Phys, 1966, 6: 1–50

    Article  Google Scholar 

  19. Liu W, Dai Y H. Minimization algorithms based on supervisor and searcher cooperation. J Optim Theory Appl, 2001, 111: 359–379

    Article  MathSciNet  MATH  Google Scholar 

  20. Nagurney A, Ramanujam P. Transportation network policy modeling with goal targets and generalized penalty functions. Transp Sci, 1996, 30: 3–13

    Article  MATH  Google Scholar 

  21. Pang J S, Fukushima M. Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput Manag Sci, 2005, 2: 21–56

    Article  MathSciNet  MATH  Google Scholar 

  22. Robbins H, Siegmund D. A convergence theorem for non negative almost super-martingales and some applications. In: Rustagi J, ed. Optimizing Methods in Statistics. New York: Academic Press, 1971, 235–257

    Google Scholar 

  23. Zhang H C, Hager W W. PACBB: a projected adaptive cyclic Barzilai-Borwein method for box constrained optimization. In: Hager W W, Huang S J, Pardalos P M, Prokopyev O A, eds. Multiscale Optimization Method and Applications: Nonconvex Optimization and Its Applications. New York: Springer, 2006, 387–392

    Chapter  Google Scholar 

Download references

Acknowledgements

The second author was supported in part by the National Natural Science Foundation of China (Grant Nos. 11871279, 11571178), and the third author was supported in part by the National Natural Science Foundation of China (Grant Nos. 11625105, 11431002).

Author information

Authors and Affiliations

  1. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Xiaomei Dong & Xingju Cai

  2. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China

    Deren Han

Authors
  1. Xiaomei Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xingju Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deren Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deren Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, X., Cai, X. & Han, D. Prediction-correction method with BB step sizes. Front. Math. China 13, 1325–1340 (2018). https://doi.org/10.1007/s11464-018-0739-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-018-0739-3

Keywords

MSC

Search

Navigation