Skip to main content
SpringerLink
Log in

A FPTAS for a class of linear multiplicative problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper we consider the problem of minimizing the product of two affine functions over a polyhedral set. An approximation algorithm is proposed and it is proved that it is a Fully Polynomial Time Approximation Scheme. We will also present and computationally investigate an exact version of the algorithm.

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. Benson, H.P.: An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming. J. Glob. Optim. 15, 315–342 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Benson, H.P., Boger, G.M.: Multiplicative programming problems: analysis and efficient point search heuristic. J. Optim. Theory Appl. 94, 487–510 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benson, H.P., Boger, G.M.: Outcome-space cutting-plane algorithm for linear multiplicative programming. J. Optim. Theory Appl. 104, 301–332 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Falk, J.E., Palocsas, S.W.: Image space analysis of generalized fractional programs. J. Glob. Optim. 4, 63–88 (1994)

    Article  MATH  Google Scholar 

  5. Gao, Y., Xu, C., Yang, Y.: An outcome-space finite algorithm for solving linear multiplicative programming. Appl. Math. Comput. 179, 494–505 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kern, W., Woeginger, G.: Quadratic programming and combinatorial minimum weight product problems. Math. Program. 110, 641–649 (2007)

    Article  MathSciNet  Google Scholar 

  7. Konno, H., Fukaishi, K.: A branch-and-bound algorithm for solving low rank linear multiplicative and fractional programming problems. J. Glob. Optim. 18, 283–299 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Konno, H., Kuno, T.: Linear multiplicative programming. Math. Program. 56, 51–64 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Konno, H., Yajima, Y., Matsui, T.: Parametric simplex algorithms for solving a special class of non-convex minimization problems. J. Glob. Optim. 1, 65–81 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Konno, H., Kuno, T., Yajima, Y.: Parametric simplex algorithms for a class of NP complete problems whose average number of steps is polynomial. Comput. Optim. Appl. 1, 227–239 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kuno, T.: A finite branch-and-bound algorithm for linear multiplicative programming. Comput. Optim. Appl. 20, 119–135 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kuno, T., Yajima, Y., Konno, H.: An outer approximation method for minimizing the product of several convex functions on a convex set. J. Glob. Optim. 3, 325–335 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu, X.J., Umegaki, T., Yamamoto, Y.: Heuristic methods for linear multiplicative programming. J. Glob. Optim. 15, 433–447 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Matsui, T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9, 113–119 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)

    MATH  Google Scholar 

  16. Pardalos, P.M.: Polynomial time algorithms for some classes of constrained quadratic problems. Optimization 21, 843–853 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–138 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ryoo, H.S., Sahinidis, N.V.: Global optimization of multiplicative programs. J. Glob. Optim. 26, 387–418 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schaible, S., Sodini, C.: Finite algorithm for generalized linear multiplicative programming. J. Optim. Theory Appl. 87, 41–55 (1995)

    Article  MathSciNet  Google Scholar 

  20. Vavasis, S.A.: Approximation algorithms for indefinite quadratic programming. Math. Program. 57, 279–311 (1992)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università di Torino, Corso Svizzera, 185, 10149, Torino, Italy

    Daniele Depetrini & Marco Locatelli

Authors
  1. Daniele Depetrini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Locatelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Locatelli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Depetrini, D., Locatelli, M. A FPTAS for a class of linear multiplicative problems. Comput Optim Appl 44, 275–288 (2009). https://doi.org/10.1007/s10589-007-9156-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9156-3

Keywords

Search

Navigation