Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems

  • Published:
Zeitschrift für Operations Research Aims and scope Submit manuscript

Abstract

Given a setΩ ofR n and a functionf fromΩ intoR n we consider a problem of finding a pointx * inΩ such that(x−x *)t f(x *)≽0 holds for every pointx inΩ. This problem is called the stationary point problem and the pointx * is called a stationary point. We present a variable dimension algorithm for solving the stationary point problem with an affine functionf on a polytopeΩ defined by constraints of linear equations and inequalities. We propose a system of equations whose solution set contains a piecewise linear path connecting a trivial starting point inΩ with a stationary point. The path can be followed by solving a series of linear programs which inherit the structure of constraints ofΩ. The linear programs are solved efficiently with the Dantzig-Wolfe decomposition method by exploiting fully the structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Dafermos S (1980) Traffic equilibrium and variational inequalities. Transportation Science 14:42–54

    Google Scholar 

  2. Eaves BC (1978) A short course in solving equations with PL homotopies. SIAM-AMS Proceeding 9:73–143

    Google Scholar 

  3. van den Elzen AH, Talman AJJ (to appear) A procedure for finding Nash equilibria in bimatrix games. Z Operat Res

  4. Hearn DW, Lawphongpanich S, Ventura JA (1985) Finiteness in restricted simplicial decomposition. Operat Res Letters 4:125–130

    Google Scholar 

  5. von Hohenbalken B (1975) A finite algorithm to maximize certain pseudoconcave functions on polytopes. Math Progr 8:189–206

    Google Scholar 

  6. von Hohenbalken B (1977) Simplicial decomposition in nonlinear programming algorithms. Math Progr 13:49–68

    Google Scholar 

  7. Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, New York

    Google Scholar 

  8. Kojima M (1980) An introduction to variable dimension algorithms for solving systems of equations. In: Allgower EL, Glashoff K, Peitgen H-O (eds) Numerical solution of nonlinear equations. Lecture Notes in Math 878. Springer, Berlin, pp 199–237

    Google Scholar 

  9. Kojima M, Kaneko I (1978) A unified parametric quadratic programming solutions to some stochastic linear programming models. Technical Summary Report No 1893. University of Wisconsin-Madison

  10. van der Laan G, Talman AJJ (1979) A restart algorithm for computing fixed points without an extra dimension. Math Progr 17:14–84

    Google Scholar 

  11. Lawphongpanich S, Hearn DH (1984) Simplicial decomposition of the asymmetric traffic assignment problem. Transportation Research 18B:123–133

    Google Scholar 

  12. Markowitz HM (1987) Mean-variance analysis in portfolio choice and capital market. Basil Blackwell, New York

    Google Scholar 

  13. Nguen S, Dupuis C (1984) An efficient method for computing traffic equilibria in networks with asymmetric transportation costs. Transportation Sci 18:184–202

    Google Scholar 

  14. Pang JS, Chan D (1982) Iterative methods for variational and complementarity problem. Math Progr 24:284–313

    Google Scholar 

  15. Pang JS, Yu C-S (1984) Linearized simplicial decomposition methods for computing traffic equilibria on networks. Networks 14:427–438

    Google Scholar 

  16. Sacher RS (1980) An algorithm for large scale convex quadratic programming. Math Progr 18:16–30

    Google Scholar 

  17. Shetty CM, Mohammed Ben Daya (1988) A decomposition procedure for convex quadratic programs. Naval Research Logistics 35:111–118

    Google Scholar 

  18. Smith MJ (1979) The existence, uniqueness and stability of traffic equilibria. Transportation Res 13B:295–304

    Google Scholar 

  19. Stoer J, Witzgall C (1978) Convexity and optimization in finite dimensions I. Springer, Berlin

    Google Scholar 

  20. Talman AJJ, Yamamoto Y (1989) A simplicial algorithm for stationary point problems. Math Operat Res 14:383–399

    Google Scholar 

  21. Yamamoto Y (1987) A path following algorithm for stationary point problems. J Operat Res Soc Japan 30:181–198

    Google Scholar 

  22. Yamamoto Y (1989) Fixed point algorithms for stationary point problems. In: Iri M, Tanabe K (eds) Mathematical programming, recent developments and applications. Kluwer Academic Publ, Dordrecht, pp 283–307

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Part of this research was carried out when the first author was supported by the Center for Economic Research, Tilburg University, The Netherlands and the third author was supported by the Alexander von Humboldt-Foundation, Federal Republic of Germany.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dai, Y., Sekitani, K. & Yamamoto, Y. A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems. ZOR Zeitschrift für Operations Research Methods and Models of Operations Research 36, 23–53 (1992). https://doi.org/10.1007/BF01541030

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01541030

Key words