Skip to main content
SpringerLink
Log in

A vector linear programming approach for certain global optimization problems

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

Abstract

Global optimization problems with a quasi-concave objective function and linear constraints are studied. We point out that various other classes of global optimization problems can be expressed in this way. We present two algorithms, which can be seen as slight modifications of Benson-type algorithms for multiple objective linear programs (MOLP). The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems. This paper generalizes results of Schulz and Mittal (Math Program 141(1–2):103–120, 2013) on quasi-concave problems and Shao and Ehrgott (Optimization 65(2):415–431, 2016) on multiplicative linear programs. Furthermore, it improves results of Löhne and Wagner (J Glob Optim 69(2):369–385, 2017) on minimizing the difference \(f=g-h\) of two convex functions g, h where either g or h is polyhedral. Numerical examples are given and the results are compared with the global optimization software BARON.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim. 13(1), 1–24 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  3. Ehrgott, M., Löhne, A., Shao, L.: A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming. J. Glob. Optim. 52(4), 757–778 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ferrer, A.: Applying global optimization to a problem in short-term hydrothermal scheduling. In: Generalized Convexity, Generalized Monotonicity and Applications, volume 77 of Nonconvex Optimization Application, pp. 263–285. Springer, New York (2005)

  5. Ferrer, A., Bagirov, A., Beliakov, G.: Solving DC programs using the cutting angle method. J. Glob. Optim. 61(1), 71–89 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fülöp, J.: On the equivalence between a linear bilevel programming problem and linear optimization over the efficient set. Technical report. Working Paper 93-1. Laboratory of Operations Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences, Budapest (1993)

  7. Hamel, A.H., Löhne, A., Rudloff, B.: Benson type algorithms for linear vector optimization and applications. J. Glob. Optim. 59(4), 811–836 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. SIAM J. Optim. 19(2), 836–845 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization, volume 2 of Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (1995)

  10. Horst, R., Tuy, H.: Global Optimization, second edn. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  11. Löhne, A.: Vector Optimization with Infimum and Supremum. Vector Optimization. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  12. Löhne, A., Wagner, A.: Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver. J. Glob. Optim. 69(2), 369–385 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Löhne, A., Weißing, B.: Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming. Math. Methods Oper. Res. 84(2), 411–426 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Löhne, A., Weißing, B., Ciripoi, D.: Bensolve tools, 2014–2017. Bensolve interface for Gnu Octave/Matlab. http://tools.bensolve.org

  15. Majthay, A., Whinston, A.: Quasi-concave minimization subject to linear constraints. Discrete Math. 9, 35–59 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mittal, S., Schulz, A.S.: An FPTAS for optimizing a class of low-rank functions over a polytope. Math. Program. 141(1–2), 103–120 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sahinidis, N.V.: BARON 14.3.1: Global optimization of mixed-integer nonlinear programs, user’s manual (2014). http://www.minlp.com/downloads/docs/baron%20manual.pdf

  18. Shao, L., Ehrgott, M.: Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes. Optimization 65(2), 415–431 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tuy, H., Tam, B.T.: An efficient solution method for rank two quasiconcave minimization problems. Optimization 24(1–2), 43–56 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Friedrich Schiller University Jena, 07737, Jena, Germany

    Daniel Ciripoi, Andreas Löhne & Benjamin Weißing

Authors
  1. Daniel Ciripoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Löhne
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benjamin Weißing
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreas Löhne.

Additional information

This research was supported by the German Research Foundation (DFG) Grant Number LO–1379/7–1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ciripoi, D., Löhne, A. & Weißing, B. A vector linear programming approach for certain global optimization problems. J Glob Optim 72, 347–372 (2018). https://doi.org/10.1007/s10898-018-0627-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-018-0627-0

Keywords

Mathematics Subject Classification

Search

Navigation