Skip to main content
SpringerLink
Log in

Asymmetric variational inequality problems over product sets: Applications and iterative methods

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper, we (i) describe how several equilibrium problems can be uniformly modelled by a finite-dimensional asymmetric variational inequality defined over a Cartesian product of sets, and (ii) investigate the local and global convergence of various iterative methods for solving such a variational inequality problem. Because of the special Cartesian product structure, these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions. The resulting decomposition schemes often have a natural interpretation as some adjustment processes.

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. H.Z. Aashtiani, “The multi-modal traffic assignment problem”, Ph.D. dissertation, Alfred P. Sloan School of Management, Massachusetts Institute of Technology (May, 1979).

  2. H.Z. Aashtiani and T.L. Magnanti, “Equilibria on a congested transportation network”,SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226.

    MATH  MathSciNet  Google Scholar 

  3. B.H. Ahn,Computation of market equilibria for policy analysis: The project independence evaluation system (PIES) approach (Garland Publishing Inc., New York 1979).

    Google Scholar 

  4. B.H. Ahn and W.W. Hogan, “On convergence of the PIES algorithm for computing equilibria”,Operations Research 30 (1982) 281–300.

    MATH  MathSciNet  Google Scholar 

  5. R.L. Asmuth, “Traffic network equilibria”, Technical Report SOL 78-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University (January, 1978).

  6. R.L. Asmuth, B.C. Eaves and E.L. Peterson, “Computing economic equilibria on affine networks with Lemke’s algorithm”,Mathematics of Operations Research 4 (1979) 209–214.

    MATH  MathSciNet  Google Scholar 

  7. R.W. Cottle and S.G. Duvall, “A Lagrangean relaxation algorithm for the constrained matrix problem”, Technical report SOL 82-10, Systems Optimization Laboratory, Department of Operations Research, Stanford University (July 1982).

  8. S.C. Dafermos, “Traffic equilibrium and variational inequalities”,Transportation Science 14 (1980) 42–54.

    MathSciNet  Google Scholar 

  9. S.C. Dafermos, “Relaxation algorithms for the general asymmetric traffic equlibrium problem”,Transportation Science 16 (1982) 231–240.

    MathSciNet  Google Scholar 

  10. S.C. Dafermos, “An iterative scheme for variation inequalities”,Mathematical Programming 26 (1983) 40–47.

    MATH  MathSciNet  Google Scholar 

  11. S.C. Fang and E.L. Peterson, “Economic equilibria on networks”, Mathematics Research Report 80-13, Department of Mathematics, University of Maryland, Baltimore County (1980).

    Google Scholar 

  12. M. Florian, “Asymmetric variable demand multi-modal traffic equilibrium problems: Existence, uniqueness and a solution algorithm”, Publication No. 152, Centre de Recherche sur les Transports, Université de Montréal (October 1979).

  13. M. Florian and H. Spiess, “The convergence of diagnoalization algorithms for asymmetric network equilibrium problems”,Transportation Research 16B (1982) 477–483.

    MathSciNet  Google Scholar 

  14. T.L. Friesz, R.L. Tobin, T.E. Smith and P.T. Harker, “A nonlinear complementarity formulation and solution procedure for the general derived demand network equilibrium problem”,Journal of Regional Science 23 (1983) 337–359.

    Article  Google Scholar 

  15. D. Gabay and H. Moulin, “On the uniqueness and stability of Nash-equilibria in noncooperative games”, in A. Bensoussan, P. Kleindorfer and C.S. Tapiero, eds.,Applied stochastic control in econometrics and management science (North-Holland, Amsterdam 1980) 271–293.

    Google Scholar 

  16. P.T. Harker, “A variational inequality aproach for the determination of oligopolistic market equilibrium”, to appear inMathematical Programming.

  17. W.W. Hogan, “Project independence evaluation system: Structure and algorithms”,Proceedings of Symposia in Applied Mathematics 21 (1977) 121–137.

    MathSciNet  Google Scholar 

  18. S. Karamardian, “The nonlinear complementarity problem with applications, Part 2”,Journal of Optimization Theory and Applications 4 (1969) 167–181.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Karamardian, “Generalized complementarity problem”,Journal of Optimization Theory and Applications 8 (1971) 161–168.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  21. S. Lawphongpanich and D.W. Hearn, “Simplicial decomposition of the asymmetric traffic assignment problem”, to appear inTransportation Research.

  22. J.G. MacKinnon, “A technique for the solution of spatial equilibrium models”,Journal of Regional Science 16 (1976) 293–307.

    Article  Google Scholar 

  23. U. Mosco, “Dual variational inequalities”, Journal ofMathematical Analysis and Applications 40 (1972) 202–206.

    Article  MATH  MathSciNet  Google Scholar 

  24. J.M. Ortega and W.C. Rheinboldt,Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970).

    MATH  Google Scholar 

  25. J.S. Pang, “A hybrid method for the solution of some multi-commodity spatial equilibrium problems”,Management Science 27 (1981) 1142–1157.

    Article  MATH  Google Scholar 

  26. J.S. Pang, “Solution of the general multi-commodity spatial equilibrium problem by variational and complementarity methods”, to appear inJournal of Regional Science.

  27. J.S. Pang, “More results on the convergence of iterative methods for the symmetric linear complementarity problem”, to appear injournal of Optimization Theory and Applications.

  28. J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems”,Mathematical Programming 24 (1982) 284–313.

    Article  MATH  MathSciNet  Google Scholar 

  29. J.S. Pang and D. Chan, “Gauss-Seidel methods for variational inequality problems over product sets”, unpublished manuscript, School of Management, University of Texas at Dallas (1982).

  30. J.S. Pang and C.S. Yu, “Linearized simplicial decomposition methods for computing traffic equlibria on networks”, to appear inNetworks.

  31. G. Pierra, “Decomposition through formalization in a product space”,Mathematical Programming 28 (1984) 96–115.

    MATH  MathSciNet  Google Scholar 

  32. S.M. Robinson, “Generalized equations”, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical programming: The state of the art (Springer-Verlag, New York 1983).

    Google Scholar 

  33. J. Rowse, “Solving the generalized transportation problem”,Regional Science and Urban Economics 11 (1981) 57–68.

    Article  Google Scholar 

  34. M.J. Smith, “Existence, uniqueness and stability of traffic equilibria”,Transportation Research 13B (1979) 295–304.

    Google Scholar 

  35. R.L. Tobin and T.L. Friesz, “Formulating and solving the spatial price equilibrium problem with transshipment in terms of arc variables”,Journal of Regional Science 23 (1983) 187–198.

    Article  Google Scholar 

  36. J.G. Wardrop, “Some theoretical aspects of road traffic research”,Proceedings of the Institute of Civil Engineers Part 2, 1 (1952) 325–378.

    Google Scholar 

  37. W.I. Zangwill and C.B. Garcia, “Equilibrium programming: The path following approach and dynamics”,Mathematical Programming 21 (1981) 262–289.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Management, The University of Texas at Dallas, Box 688, 75080, richardson, TX, USA

    Jong-Shi Pang

Authors
  1. Jong-Shi Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was based on work supported by the National Science Foundation under grant ECS 811–4571.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pang, JS. Asymmetric variational inequality problems over product sets: Applications and iterative methods. Mathematical Programming 31, 206–219 (1985). https://doi.org/10.1007/BF02591749

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02591749

Key words

Search

Navigation