Skip to main content
SpringerLink
Log in

Decomposition Branch-and-Bound Based Algorithm for Linear Programs with Additional Multiplicative Constraints

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This article presents an algorithm for globally solving a linear program (P) that contains several additional multiterm multiplicative constraints. To our knowledge, this is the first algorithm proposed to date for globally solving Problem (P). The algorithm decomposes the problem to obtain a master problem of low rank. To solve the master problem, the algorithm uses a branch-and-bound scheme where Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems in the algorithm are ordinary linear programs. Convergence of the algorithm is shown and a solved sample problem is given.

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. H. Tuy (1998) Convex Analysis and Global Optimization Kluwer Dordrecht, Netherlands

    Google Scholar 

  2. N. V. Thoai (2002) ArticleTitleConvergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization Journal of Optimization Theory and Applications 113 165–193 Occurrence Handle10.1023/A:1014865432210

    Article  Google Scholar 

  3. T. Kuno H. Konno A. Irie (1999) ArticleTitleA Deterministic Approach to Linear Programs with Several Additional Multiplicative Constraints Computational Optimization and Applications 14 347–366 Occurrence Handle10.1023/A:1026404623838

    Article  Google Scholar 

  4. H. Konno T. Kuno (1992) ArticleTitleLinear Multiplicative Programming Mathematical Programming 56 51–64 Occurrence Handle10.1007/BF01580893

    Article  Google Scholar 

  5. H. Konno T. Kuno (1995) Multiplicative Programming Problems R. Horst P. M. Pardalos (Eds) Handbook of Global Optimization Kluwer Dordrecht, Netherlands 369–405

    Google Scholar 

  6. H. Konno C. Gao I. Saitoh (1998) ArticleTitleCutting Plane/Tabu Search Algorithms for Solving Low-Rank Concave Quadratic Programming Problems Journal of Global Optimization 13 225–240 Occurrence Handle10.1023/A:1008230825152

    Article  Google Scholar 

  7. T. Kuno T. Utsunomiya (1997) ArticleTitleA Pseudopolynomial Primal-Dual Algorithm for Globally Solving a Production-Transportation Problem Journal of Global Optimization 11 163–180 Occurrence Handle10.1023/A:1008278625289

    Article  Google Scholar 

  8. J. F. Benders (1962) ArticleTitlePartitioning Procedures for Solving Mixed-Variables Programming Problems Numerische Mathematik 4 238–252 Occurrence Handle10.1007/BF01386316

    Article  Google Scholar 

  9. C. A. Floudas V. Visweswaran (1993) ArticleTitleA Primal-Relaxed Dual Global Optimization Approach Journal of Optimization Theory and Applications 78 187–225 Occurrence Handle10.1007/BF00939667

    Article  Google Scholar 

  10. R. Horst N. V. Thoai (1996) ArticleTitleDecomposition Approach for the Global Minimization of Biconcave Functions over Polytopes Journal of Optimization Theory and Applications 88 561–583

    Google Scholar 

  11. L. D. Muu W. Oettli (1991) ArticleTitleMethod for Minimizing Convex-Concave Functions over a Convex Set Journal of Optimization Theory and Applications 70 377–384 Occurrence Handle10.1007/BF00940633

    Article  Google Scholar 

  12. N. V. Thoai (1991) ArticleTitleA Global Optimization Approach for Solving the Convex Multiplicative Programming Problem Journal of Global Optimization 1 341–357 Occurrence Handle10.1007/BF00130830

    Article  Google Scholar 

  13. H. P. Benson G. M. Boger (1997) ArticleTitleMultiplicative Programming Problems: Analysis and Efficient Point Search Heuristic Journal of Optimization Theory and Applications 94 487–510 Occurrence Handle10.1023/A:1022600232285

    Article  Google Scholar 

  14. H. P. Benson (1999) ArticleTitleAn Outcome Space Branch-and-Bound–Outer–Approximation Algorithm for Convex Multiplicative Programming Journal of Global Optimization 15 315–342 Occurrence Handle10.1023/A:1008316429329

    Article  Google Scholar 

  15. H. P. Benson G. M. Boger (2000) ArticleTitleAn Outcome Space, Cutting Plane Algorithm for Linear Multiplicative Programming Journal of Optimization Theory and Applications 104 301–322 Occurrence Handle10.1023/A:1004657629105

    Article  Google Scholar 

  16. H. P. Benson G. M. Boger (2000) Analysis of an Outcome Space Formulation of the Multiplicative Programming Problem M. Zeleny Y. Shi (Eds) New Frontiers of Decision Making for the Information Technology Era World Scientific Publishing River Edge, New Jersey 100–122

    Google Scholar 

  17. M. Dur (2001) ArticleTitleDual Bounding Procedures Lead to Convergent Branch-and-Bound Algorithms Mathematical Programming 91 117–125

    Google Scholar 

  18. M. Dur (2002) ArticleTitleA Class of Problems Where Dual Bounds Beat Underestimation Bounds Journal of Global Optimization 22 49–57 Occurrence Handle10.1023/A:1013890609372

    Article  Google Scholar 

  19. M. Dur R. Horst (1997) ArticleTitleLagrange-Duality and Partitioning Techniques in Nonconvex Global Optimization Journal of Optimization Theory and Applications 95 347–369 Occurrence Handle10.1023/A:1022687222060

    Article  Google Scholar 

  20. I. Nowak (2000) ArticleTitleDual Bounds and Optimality Cuts for All-Quadratic Programs with Convex Constraints Journal of Global Optimizatiion 18 337–356 Occurrence Handle10.1023/A:1026596100403

    Article  Google Scholar 

  21. N. V. Thoai (2000) ArticleTitleDuality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints Journal of Optimization Theory and Applications 107 331–354 Occurrence Handle10.1023/A:1026437621223

    Article  Google Scholar 

  22. R. Horst H. Tuy (1996) Global Optimization: Deterministic Approaches Springer Berlin, Germany

    Google Scholar 

  23. M. Avriel W. E. Diewert S. Schaible I. Zang (1988) Generalized Concavity Plenum New York, NY

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Professor, Department of Decision and Information Sciences, University of Florida, Gainesville, Florida

    H. P. Benson

Authors
  1. H. P. Benson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benson, H.P. Decomposition Branch-and-Bound Based Algorithm for Linear Programs with Additional Multiplicative Constraints. J Optim Theory Appl 126, 41–61 (2005). https://doi.org/10.1007/s10957-005-2655-4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-2655-4

Keywords

Search

Navigation