Advertisement

Location of a conservative hyperplane for cutting plane methods in disjoint bilinear programming

  • Xi Chen
  • Ji-hong Zhang
  • Xiao-song DingEmail author
  • Tian Yang
  • Jing-yi Qian
Original paper
  • 8 Downloads

Abstract

Although several classes of cutting plane methods for deterministically solving disjoint bilinear programming (DBLP) have been proposed, the frequently encountered computational issue regarding the generation of a suitable cut from a degenerate vertex in a pseudo-global minimizer (PGM) still remains. Among the approaches to dealing with degeneracy, the most recent one is to generate a conservative cut. Nevertheless, the computational performance of the corresponding distance-following algorithm for its location seems far from satisfactory. This paper proposes several approaches that can be utilized to efficiently locate a conservative hyperplane from a degenerate vertex in a PGM. Extensive experiments are conducted to evaluate their performance from the dimensionality as well as the degree of degeneracy. From the computational viewpoint, these new approaches can outperform the earlier developed distance-following algorithm, and thereby can be incorporated into cutting plane methods for solving DBLP.

Keywords

Cutting plane method Degeneracy removal Conservative cut 

Notes

Acknowledgements

The authors are grateful to the editor and the anonymous referee for their insightful comments to significantly improve this paper.

References

  1. 1.
    Alarie, S., Audet, C., Jaumard, B., Savard, G.: Concavity cuts for disjoint bilinear programming. Math. Program. 90(2), 373–398 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Audet, C., Hansen, P., Jaumard, B., Savard, G.: A symmetrical linear maxmin approach to disjoint bilinear programming. Math. Program. 85(3), 573–592 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Konno, H.: Bilinear programming: Part II. Application of bilinear programming. Tech. Rept. No. 71–10, Department of Operations Research, Stanford University, Stanford, Calif. (1971)Google Scholar
  4. 4.
    Konno, H.: A cutting plane algorithm for solving bilinear programs. Math. Program. 11(1), 14–27 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Nahapetyan, A.: Bilinear programming: applications in the supply chain management. In: Encyclopedia of optimization, pp 282–288. Springer (2009)Google Scholar
  6. 6.
    Porembski, M.: On the hierarchy of \(\gamma \)-valid cuts in global optimization. Naval Res. Logist. 55(1), 1–15 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rebennack, S., Nahapetyan, A., Pardalos, P.M.: Bilinear modeling solution approach for fixed charge network flow problems. Optim. Lett. 3(3), 347–355 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sherali, H.D., Shetty, C.M.: A finitely convergent algorithm for bilinear programming problems using polar cuts and disjunctive face cuts. Math. Program. 19(1), 14–31 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Vaish, H., Shetty, C.M.: A cutting plane algorithm for the bilinear programming problem. Naval Res. Logist. 24(1), 83–94 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, J., Chen, X., Ding, X.: Degeneracy removal in cutting plane methods for disjoint bilinear programming. Optim. Lett. 11(3), 483–495 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Xi Chen
    • 1
  • Ji-hong Zhang
    • 1
  • Xiao-song Ding
    • 1
    Email author
  • Tian Yang
    • 2
  • Jing-yi Qian
    • 3
  1. 1.International Business SchoolBeijing Foreign Studies UniversityBeijingPeople’s Republic of China
  2. 2.School of Computer Science and TechnologyBeijing Institute of TechnologyBeijingPeople’s Republic of China
  3. 3.High School Affiliated to Shanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

Personalised recommendations