Variational Inequalities and Continuous Optimal Control

  • Bin Ran
  • David Boyce


Variational inequality (VI) problems are being used extensively in formulating economic and engineering problems in new ways that lead to tractable solution algorithms. Applications in transportation engineering are relatively recent, beginning about 1980. Because of its general capability to formulate transportation problems, VI has received increasing attention from transportation network modelers during the last decade. In this chapter, we introduce the basic concepts of variational inequality theory which is capable of formulating and analyzing more general problems than the constrained optimization approach. In Section 2.1, we first define variational inequality problems for both static and dynamic problems. We then introduce some fundamental definitions, along with qualitative results for variational inequality problems, such as conditions for existence and uniqueness of solutions. Subsequently, the relaxation method for solving a variational inequality is discussed.


Variational Inequality Optimal Control Problem Complementarity Problem Variational Inequality Problem Level Problem 
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Copyright information

© Springer-Verlag Berlin · Heidelberg 1996

Authors and Affiliations

  • Bin Ran
    • 1
  • David Boyce
    • 2
  1. 1.Department of Civil and Environmental EngineeringUniversity of Wisconsin at MadisonMadisonUSA
  2. 2.Urban Transportation CenterUniversity of Illinois at ChicagoChicagoUSA

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