Skip to main content
SpringerLink
Log in

Stochastic level-value approximation for quadratic integer convex programming

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

We propose a stochastic level value approximation method for a quadratic integer convex minimizing problem in this paper. This method applies an importance sampling technique, and make use of the cross-entropy method to update the sample density functions. We also prove the asymptotic convergence of this algorithm, and report some numerical results to illuminate its effectiveness.

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. Laurence A. Wolsey, integer programming[M]. New York: John Wiley and Sons, Inc. 1998.

    Google Scholar 

  2. Land A H, Doig A G. An automatic method for solving discrete problems[J]. Econometrica, 1960, 28(3):497–520.

    Article  MATH  MathSciNet  Google Scholar 

  3. Dakin R J. A tree search algorithm for mixed integer programming problems[J]. Computer Journal, 1965, 8(3):250–255.

    Article  MATH  MathSciNet  Google Scholar 

  4. Dantzig G B, Fulkerson D R, Johnson S. Solution of a large scale traveling salesman problem[J]. Operations Research, 1954, 2(4):393–410.

    Article  MathSciNet  Google Scholar 

  5. Gomory R E. An algorithm for the mixed integer problem[R]. RM-2597, The Rand Corporation, 1960.

  6. Derpich I, Vera J R. Improving the efficiency of the branch and bound algorithm for integer programming based on “flatness” information[J]. European Journal of Operational Research, 2006, 174(1):92–101.

    Article  MATH  MathSciNet  Google Scholar 

  7. Hua Zhongsheng, Huang Feihua. An effective genetic algorithm approach to large scale mixed integer programming problems[J]. Applied Mathematics and Computation, 2006, 174(2):897–909.

    Article  MATH  MathSciNet  Google Scholar 

  8. Sandro Bosio, Giovanni Righini. Computational approaches to a combinatorial optimization problem arising from text classification[J]. Computers and Operations Research, 2007, 34(7):1910–1928.

    Article  MATH  Google Scholar 

  9. Arostegui Marvin A Jr, Kadipasaoglu Sukran N, Khumawala Basheer M. An empirical comparison of tabu search, simulated annealing, and genetic algorithms for facilities location problems[J]. International Journal of Production Economics, 2006, 103(2):742–754.

    Article  Google Scholar 

  10. Dino Ahr and Gerhard Reinelt. A tabu search algorithm for the min-max k-Chinese postman problem[J]. Computers and Operations Research, 2006, 33(12):3403–3422.

    Article  MATH  MathSciNet  Google Scholar 

  11. Mokhtar S B, Hanif D S and Shetty C M. Nonlinear programming, theory and algorithms[M]. Canada: John Wiley and Sons, Inc. 1993, 199–233.

    Google Scholar 

  12. De Boer P-T, Kroese D P, Mannor S, Rubinstein R Y. A tutorial on the cross-entropy method[J]. Annals of Operations Research, 2005, 134(1):19–67.

    Article  MATH  MathSciNet  Google Scholar 

  13. Wu Donghua, Yu Wuyang, Tian Weiwen. A level-value estimation method for solving global optimization[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(7):1001–1010. DOI 10.1007/s 10483-006-0717-1

    Article  MATH  MathSciNet  Google Scholar 

  14. Peng Zheng, Wu Donghua, Tian Weiwen. A level-value estimation method for solving constrained global optimization[J]. Mathematical Numerica Sinica, 2007, 29(3):293–304 (in Chinese).

    MATH  MathSciNet  Google Scholar 

  15. Rubinstein R Y. The cross-entropy method for combinatorial and continuous optimization[J]. Methodology and Computing in Applied Probability, 1999, 1(2):127–190.

    Article  MATH  MathSciNet  Google Scholar 

  16. Ross, Sheldon M Simulation[M]. 3rd Edition. Beijing: Posts and Telecom Press in China, 2006, 45–47.

    Google Scholar 

  17. Wan Guocheng, Tian Xiang, Ren Zhen. A solution to King of nonlinear integer programming problem with computer[J]. Computer Engineering and Applications. 2002, 38(16): 80–83 (in Chinese).

    Google Scholar 

  18. LIN Jin, ZHU Wenxing. A hybrid ant colony system for the convex ineger programing problem[J]. Journal of Fuzhou University (Natural Science), 1999, 27(6):5–9 (in Chinese).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China

    Zheng Peng  (彭拯) & Dong-hua Wu  (邬冬华)

  2. Department of mathematics, Hunan Institute of Science and Technology, Yueyang, 414006, Hunan Province, P. R. China

    Zheng Peng  (彭拯)

Authors
  1. Zheng Peng  (彭拯)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dong-hua Wu  (邬冬华)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dong-hua Wu  (邬冬华).

Additional information

Communicated by LIU Zeng-rong

Project supported by the National Natural Science Foundation of China (No. 10671117), Shanghai Leading Academic Discipline Project (No. J050101) and the Youth Science Foundation of Hunan Education Department of China (No. 06B037)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peng, Z., Wu, Dh. Stochastic level-value approximation for quadratic integer convex programming. Appl. Math. Mech.-Engl. Ed. 29, 801–809 (2008). https://doi.org/10.1007/s10483-008-0611-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-008-0611-y

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation