Skip to main content
SpringerLink
Log in

Dynamic Competition on Networks: Differential Variational Inequalities, Limited Warfare and Internet Vulnerability

  • Chapter
Network Science, Nonlinear Science and Infrastructure Systems

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 102))

Abstract

In this paper we are concerned with a class of mathematical problems that arise when modelling dynamic competition among conflicting agents that may or may not reach an equilibrium. We call this class of problems differential variational inequalities. We are further interested in the application of models belonging to this class of problems to the study of (1) policy constrained, limited warfareand (2) Internet vulnerability.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • G. Anandalingam and T.L. Friesz (1992), Hierarchical Optimization: An Introduction, Annals of Operations Research, 34, pp. 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Anandalingam and T.L. Friesz (eds) (1992), Hierarchical Mathematical Programming, Baltzer AG Scientific Publishing Co.

    Google Scholar 

  • J.P. Aubin and A. Cellina (1984), Differential Inclusions, Springer-Verlag, Berlin, 342 pp., 1984.

    MATH  Google Scholar 

  • C. Baiocchi and A. Capelo (1984), Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, Wiley.

    Google Scholar 

  • T. Basar and G.J. Olsder (1998), Dynamic Noncooperative Game Theory, nd Edition, SIAM, Philadelphia.

    Google Scholar 

  • D. Bernstein and S.A. Gabriel (1997), “Solving the Nonadditive Traffic Equilibrium Problem,” in Lecture Notes in Economics and Mathematical Systems: Network Optimization, P.M. Pardalos, D.W. Hearn, W.W. Hager, eds., Springer, 72–102.

    Google Scholar 

  • M. Ferris and J.S. Pang (1997), Engineering and economic applications of complementarity problems, SIAM Review, 39, 669–713.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Friedman (1982), Variational Principles and Free Boundary Problems, Krieger.

    Google Scholar 

  • T.L. Friesz and S. Shah (2001a), An Overview of Nontraditional Formulations of Static and Dynamic Equilibrium Network Design, Transportation Research, 35(1).

    Google Scholar 

  • T.L. Friesz and S. Shah (2001b), Foundations of a Theory of Disequilibrium Design, paper 10 in Transportation Networks: Recent Methodological Advances (M. Bell, Editor), Elsevier Science Publishers, pp. 143–162, 1999.

    Google Scholar 

  • T.L. Friesz, S. Shah, D. Bernstein (1999), Disequilibrium Network Design: A New Paradigm for Transportation Planning and Control, paper 6 in Network Infrastructure and the Urban Environment (L. Lundquist et. al, editors), Springer-Verlag, pp. 99–111.

    Google Scholar 

  • T.L. Friesz, Z. Suo, D. Bernstein and R. Tobin (2001), Differential Variational Inequalities and Dynamic Network Equilibrium, Networks and Spatial Economics, 1(3).

    Google Scholar 

  • T.L. Friesz, et al.(1992), Simulated Annealing Methods for Network Design Problems with Variational Inequality Constraints, Transportation Science, 26(1), pp. 18–26.

    MATH  MathSciNet  Google Scholar 

  • T.L. Friesz, et al.(1993), The Multiobjective Equilibrium Network Design Problem Revisited: A Simulated Annealing Approach, European Journal of Operational Research, 65, pp. 44–57.

    Article  MATH  Google Scholar 

  • S.A. Gabriel (1998), An NE/SQP Method for the Bounded Nonlinear Complementarity Problem, Journal for Optimization Theory and Applications, 97, 493–506.

    Article  MATH  MathSciNet  Google Scholar 

  • S.A. Gabriel (1998), A Hybrid Smoothing Method for the Mixed Nonlinear Complementarity Problem, Computational Optimization and Applications, 9, 153–173.

    Article  MATH  MathSciNet  Google Scholar 

  • S.A. Gabriel and D. Bernstein (1997), The Traffic Equilibriuim Problem with Nonadditive Costs, Transportation Science, 31, 337–348.

    Article  MATH  Google Scholar 

  • S.A. Gabriel and D. Bernstein (2000), Nonadditive Shortest Paths: Subproblems in Multi-Agent Competitive Network Models, Computational and Mathematical Organization Theory, 6, 29–45.

    Article  Google Scholar 

  • S.A. Gabriel and J.S. Pang (1992), An Inexact NE/SQP Method for Solving the Nonlinear Complementarity Problem, Computational Optimization and Applications, 1, 67–91.

    Article  MATH  MathSciNet  Google Scholar 

  • S.A. Gabriel and J.S. Pang (1994), A Trust Region Method for Constrained Nonsmooth Equations, in Large-Scale Optimization: State of the Art, edited by W.W. Hager, D.W. Hearn, and P. Pardalos, Kluwer Academic Publishers, 159–186.

    Google Scholar 

  • S.A. Gabriel and J.J. More (1997), Smoothing of Mixed Complementarity Problems, in Proceedings of the International Conference on Complementarity Problems, The Johns Hopkins University, Baltimore, Maryland, 105–116.

    Google Scholar 

  • S.A. Gabriel, A.S. Kydes, P. Whitman (2001), The National Energy Modeling System: A Large-Scale Energy-Economic Equilibrium Model, Theory and Computational Results, Vol. 29, Operations Research, 14–25.

    Google Scholar 

  • R.J. Gibbens and Kelly, Resource Pricing and the Evolution of Congestion Control, Automatica, 35, 1999

    Google Scholar 

  • P.T. Harker (1985), Investigating the Use of the Core as a Solution Concept in Spatial Price Equilibrium Games, Spatial Price Equilibrium: Advances in Theory, Computation and Application, Springer-Verlag (P.T. Harker, editor), 41–72.

    Google Scholar 

  • P.T. Harker and J.S. Pang (1990), Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48, 161–220.

    Article  MATH  MathSciNet  Google Scholar 

  • R. Isaacs (1965), Differential Games, First Edition, Wiley.

    Google Scholar 

  • F.P. Kelly (1997) Charging and Rate Control for Elastic Traffic, European Transactions on Telecommunications, 8, pp. 33–37.

    Google Scholar 

  • F.P. Kelly (2000), Mathematical Modelling of the Internet, working paper, Statistical Laboratory, University of Cambridge.

    Google Scholar 

  • D. Kinderlehrer and G. Stampacchia (1980), An Introduction to Variational Inequalities and Their Applications, Academic Press.

    Google Scholar 

  • V.K. Le and K. Schmitt (1997), Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems, Springer Verlag.

    Google Scholar 

  • Z.Q. Luo, AJ.S. Pang and D. Ralph (1996), Mathematical Programs with Equilibrium Constraints, Cambridge University Press.

    Google Scholar 

  • D.G. Luenberger, Optimization by Vector Space Methods (1969), Wiley.

    Google Scholar 

  • R. Mantegna and H. Stanley (2000), An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press.

    Google Scholar 

  • M. Minoux, Mathematical Programming: Theory and Algorithms (1986), Wiley.

    Google Scholar 

  • J.S. Pang and S.A. Gabriel (1993), NE/SQP: a Robust Algorithm for the Nonlinear Complementarity Problem, Mathematical Programming, 60, 295–337.

    Article  MathSciNet  Google Scholar 

  • P.A. Samuelson (1947), Foundations of Economic Analysis, Harvard University Press

    Google Scholar 

  • T.E. Smith, T.L. Friesz et al. (1997), A Comparison of Two Minimum Norm Projective Dynamic Systems and Their Relationship to Variational Inequalities, Complementarity and Variational Problems (M. Ferris and J.-S. Pang, editors), SIAM, pp. 405–424.

    Google Scholar 

  • P. Thornton (1995), A Hierarchical Mathematical Programming Model of the U.S. Army Promotion and Assignment Process, Ph.D. dissertation, George Mason University.

    Google Scholar 

  • H.R. Varian (1984), Microeconomic Analysis, 2nd Edition, Norton.

    Google Scholar 

  • W. Willinger and V. Paxson (1998), Where Mathematics Meets the Internet, Notices of the American Mathematical Society, 45, pp. 961–970 (www.ams.org/notices/).

  • P. Wilmott, J. Dewynne and S. Howison (1993), Option Pricing: Mathematical Models and Computation, Oxford Financial Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Pennsylvania State University, University Park, PA, 16802, USA

    Terry L. Friesz

Authors
  1. Terry L. Friesz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Terry L. Friesz .

Editor information

Editors and Affiliations

  1. Pennsylvania State University, University Park, PA, USA

    Terry L. Friesz

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Friesz, T.L. (2007). Dynamic Competition on Networks: Differential Variational Inequalities, Limited Warfare and Internet Vulnerability. In: Friesz, T.L. (eds) Network Science, Nonlinear Science and Infrastructure Systems. International Series in Operations Research & Management Science, vol 102. Springer, Boston, MA. https://doi.org/10.1007/0-387-71134-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation