Efficient minimization over products of simplices and its application to nonlinear multicommodity network problems

  • Athanasia Karakitsiou
  • Athanasia Mavrommati
  • Athanasios Migdalas

Abstract

Many problems in portfolio selection, in traffic planning and in computer communication networks can be formulated as nonlinear problems involving minimization of a nonlinear function over simplices. Based on the concepts of regularization and partial linearization, we propose an efficient solution technique for such programs, prove its global convergence and discuss the possibilities for parallel implementation. We provide, as an application, an algorithm for traffic assignment which outperforms previous state-of-the-art codes.

Keywords

network problem linearization solution techniques 

References

  1. Bazaraa M.S., Jarvis J.J., and Sherali H.D. (1979) Nonlinear Programming. Theory and Algorithms. John Wiley & Sons, NY.Google Scholar
  2. Bazaraa M.S., Jarvis J.J., and Sherali H.D. (1990) Linear Programming and Network Flows. John Wiley & Sons, NY.Google Scholar
  3. Bertsekas D.P and Tsitsiklis J.N. (1989). Parallel and Distributed Computation. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  4. Bertsekas, D.P and Gallager R. (1992).Data Networks. Prentice-Hall, Englewood Cliffs, NJ, 1992.Google Scholar
  5. Brucker P. (1984)An O(n) algorithm for quadratic knapsack problems. Operations Research Letters vol. 3, 163–166.CrossRefGoogle Scholar
  6. Damberg O. and Migdalas A. (1997). Distributed disaggregate simplicial decomposition — A parallel algorithm for traffic assignment. In D. Hearn et al, editors,Network Flows. Springer-Verlag.Lecture Notes in Economics and Mathematical Systems, 450, 172–193.Google Scholar
  7. Dussault, J.P, Ferland J.A., and Lemaire B. (1986) Convex quadratic programming with one constraint and bounded variables. Mathematical Programming vol. 36, 90–104.CrossRefGoogle Scholar
  8. Frank M. and Wolfe P. (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly vol. 3, 95–110.CrossRefGoogle Scholar
  9. Hearn D.W, Lawphongpanich, S., and Ventura, J.A. (1985). Finiteness in restricted simplicial decomposition. Operations Research Letters vol. 4, 125–130.CrossRefGoogle Scholar
  10. Hearn, D.W., Lawphongpanich, S. and Ventura, J.A. (1987) Restricted simplicial decomposition: computation and extensions. Mathematical Programming Study vol. 3, 99–118.Google Scholar
  11. Helgason K., Kennington J., and Lall H. (1980). A polynomially bounded algorithm for a singly constrained quadratic program. Mathematical Programming vol. 18, 99–118CrossRefGoogle Scholar
  12. INRIA-ENPC. Free Scientific Software Package Scilab. 1989–2003 copyright by INRIA-ENPC. Available from http://www.scilab.org.Google Scholar
  13. Larsson T. and Migdalas A. (1990). An algorithm for nonlinear programs over Cartesian product sets Optimization vol. 21, 535–542.Google Scholar
  14. Larsson T., and Patriksson M. (1992). Simplicial decomposition with disaggregated representation for the traffic assignment problem Transportation Science vol. 26, 4–17.CrossRefGoogle Scholar
  15. Larsson T., Migdalas A., and Patriksson M. A partial linearization method for the traffic assignment problem Optimization vol. 28, 47–61.Google Scholar
  16. Lawphongpanich S., and Hearn D.W (1984). Simplicial decomposition of the asymmetric traffic assignment problem. Transportation Research vol. 18B, 123–133.Google Scholar
  17. Lotito, P. Mancinelli E., Quadrat J.P, and Wynter L.(2003). CiudadSim2: Traffic Toolbox for Scilab. Available from http://www.scilab.org.Google Scholar
  18. Migdalas A. (1994). A regularization of the Frank-Wolfe algorithm and unification of certain nonlinear programming methods. Mathematical Programming vol. 65, 331–345.CrossRefGoogle Scholar
  19. Pang J.S. (1980) A new and efficient algorithm for a class of portfolio selection problems. Operations Research vol. 28, 754–767.Google Scholar
  20. Pardalos P.M, and Kovoor N. (1990). An Algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds. Mathematical Programming vol. 46, 321–328.CrossRefGoogle Scholar
  21. Patriksson M. (1993).A Unified Framework of Descent Algorithms for Nonlinear Programs and Variational Inequalities. Linkoping Studies in Science and Technology. Dissertation No. 308, Division of Optimization, Department of Mathematics, Linkoping Institute of Technology, S-581 83 Linkoping, Sweden.Google Scholar
  22. Pschenichny B.N., and Danilin Y.M. (1982). Numerical Methods in Extremal Problems. Mir Publishers. Translated from the Russian by V. Zhitomirsky.Google Scholar
  23. von Hohenbalken B. (1977). Simplicial decomposition in nonlinear programming algorithms. Mathematical Programming vol. 13, 49–68.CrossRefGoogle Scholar
  24. Wu J.H. (1991). A Study of Monotone Variational Inequalities and their Application to Network Equilibrium Problems. PhD Thesis, Centre de Rechereche sur les Transports, Universitι de Montrιal, Montreal, Canada.Google Scholar

Copyright information

© Hellenic Operational Research Society 2004

Authors and Affiliations

  • Athanasia Karakitsiou
    • 1
  • Athanasia Mavrommati
    • 1
  • Athanasios Migdalas
    • 1
  1. 1.DSS Laboratory Department of Production Engineering and ManagementTechnical University of CreteChaniaGreece

Personalised recommendations