Advertisement

Mathematical Methods of Operations Research

, Volume 80, Issue 1, pp 83–97 | Cite as

On DC optimization algorithms for solving minmax flow problems

  • Le Dung MuuEmail author
  • Le Quang Thuy
Original Article

Abstract

We formulate minmax flow problems as a DC optimization problem. We then apply a DC primal-dual algorithm to solve the resulting problem. The obtained computational results show that the proposed algorithm is efficient thanks to particular structures of the minmax flow problems.

Keywords

Minmax flow problem Smooth DC optimization Regularization 

Notes

Acknowledgments

We would like to thank the Associate Editor and the anonymous referees for their useful suggestions, comments and remarks on the paper, which helped us very much in revising the paper.

References

  1. An LTH, Tao PD, Muu LD (1996) Numerical solution for optimization over the efficient set by DC optimization algorithms. Oper Res Lett 19:117–128zbMATHMathSciNetCrossRefGoogle Scholar
  2. An LTH, Tao PD (1997) Convex analysis approach to DC programming: theory, algorithms and applications. ACTA Math Vietnam 22:289–355zbMATHMathSciNetGoogle Scholar
  3. An LTH, Tao PD, Muu LD (2003) Simplically-constrained DC optimization over the efficient and weakly sets. J Optim Theory Appl 117:503–531zbMATHMathSciNetCrossRefGoogle Scholar
  4. Benson HP (1984) Optimization over the efficient set. J Math Anal Appl 98:56–580MathSciNetCrossRefGoogle Scholar
  5. Benson HP (1991) An all-linear programming relaxation algorithm for optimization over the efficient set. J Global Optim 1:83–104zbMATHMathSciNetCrossRefGoogle Scholar
  6. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  7. Gotoh J, Thoai NV, Yamamoto Y (2003) Global optimization method for solving the minimum maximal flow problem. Optim Methods Softw 18:395–415zbMATHMathSciNetCrossRefGoogle Scholar
  8. Iri M (1994) An essay in the theory of uncontrollable flows and congestion, Technical Report, Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, TRISE 94–03Google Scholar
  9. Iri M (1996) Network flow—theory and applications with practical impact. In: Dolezal J, Fidler J (eds) System modelling and optimization. Chapman and Hall, London, pp 24–36CrossRefGoogle Scholar
  10. Kim NTB, Muu LD (2002) On the projection of the efficient set and potential applications. Optimization 51:401–421zbMATHMathSciNetCrossRefGoogle Scholar
  11. Luc LT, Muu LD (1997) A global optimization approach to optimization over the efficient set. In: Lecture notes in economics and mathematical systems, Springer, Berlin, 452, pp 183–196Google Scholar
  12. Philip J (1972) Algorithms for the vector maximization problem. Math Program 2:207–229zbMATHMathSciNetCrossRefGoogle Scholar
  13. Shi JM, Yamamoto Y (1997) A global optimization method for minimum maximal flow problem. Acta Math Vietnam 22:271–287zbMATHMathSciNetGoogle Scholar
  14. Shigeno M, Takahashi I, Yamamoto Y (2003) Minimum maximal flow problem—an optimization over the efficient set. J Glob Optim 25:425–443zbMATHMathSciNetCrossRefGoogle Scholar
  15. Tao PD (1986) Algorithms for solving a class of non convex optimization. Methods of subgradients, Fecmat Days 85, Mathematics for Optimization, Elservier Sience Publishers, B.V. NorthHollandGoogle Scholar
  16. Yamamoto Y (2002) Optimization over the efficient set: overview. J Glob Optim 22:285–317zbMATHCrossRefGoogle Scholar
  17. Zenke D (2006) Algorithms for the minimum maximal flow problem—optimization over the efficient set, Graduate School of Systems and Information Engineering, University of TsukubaGoogle Scholar
  18. Zhu DL, Marcotte P (1994) An extended descent framework for variational inequalities. J Optim Theory Appl 80:349–366zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of MathematicsVASTHanoiVietnam
  2. 2.School of Applied Mathematics and InformaticsHaNoi University of Science and TechnologyHanoiVietnam

Personalised recommendations