Generalized Equations

  • Jiři Outrata
  • Michal Kočvara
  • Jochem Zowe
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 28)


As we will see later, variational inequalities (and complementarity problems) provide a convenient and elegant tool for characterizing manifold equilibria. The aim of this chapter is to spell out how these models can be brought into the equally useful form of a generalized equation
$$ 0 \in C\left( z \right) + {N_Q}\left( z \right),$$
where C[ℝ k fℝ k ] is a continuous mapping, Q a nonempty, closed, convex subset of ℝk and N Q (z) its normal cone to Q at z; cf. Definition 2.6. Q is called the feasible set of the GE (4.1). Oftentimes, the rewriting as a “nonsmooth equation” is not only possible but very helpful. While proceeding, we also collect several basic results on existence and uniqueness needed in the later chapters. Our objective is to prepare for subsequent analysis and computations.


Variational Inequality Complementarity Problem Normal Cone Linear Complementarity Problem Positive Definiteness 
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Bibliographical notes

  1. The variational inequality problem was introduced by Hartman and Stampacchia in 1966 and subsequently expanded in several classic papers. These early studies were motivated by boundary value problems posed in the form of partial differential equations, cf. Hartman and Stampacchia, 1966 and Kinderlehrer and Stampaccia, 1980. The possibility to model various economic equilibria (Nash equilibrium, Wardrop equilibrium, etc.) as finite-dimensional VIs was recognized only much later; cf. Harker and Pang, 1990 and theGoogle Scholar
  2. references therein. The nonlinear complementarity problem first appeared in 1966 (Cottle, 1966), but only a few years later it was recognized as a special case of the VI (Karamardian, 1971). Quasivariational inequalities were introduced in the seventies by Bensoussan and Lions in connection with stochastic impulse control problems (e.g. Bensoussan and Lions, 1973). They also turned out to be very convenient for modelling various equilibria in both mechanics (Mosco, 1976; Baiocchi and Capelo, 1984) and mathematical economy (Harker, 1991).Google Scholar
  3. The generalized equations first appeared in the cited works by Robinson in the form (4.1), i.e., with the normal cone mapping. However, in the recent works on this subject no special structure of the set-valued part is assumed (Dontchev and Hager, 1994; Dontchev, 1995).Google Scholar
  4. Theorem 4.1 is the famous result from Hartman and Stampacchia, 1966 (in the finitedimensional setting) and also Theorem 4.2 comes from this paper. Proposition 4.5 is extracted from Ortega and Rheinboldt, 1970, where various monotonicity properties of an operator are related to positive definiteness or positive semi-definiteness of the respective Jacobians. The existence and uniqueness questions in LCPs are fairly well understood and the underlying theory goes far beyond the scope of this book. The interested reader is referred to Murty, 1988 or Cottle et al., 1992. Finally, Theorem 4.9 is a generalization of a result from Pang, 1981, concerning the linear implicit complementarity problem and comes from Kočvara and Outrata, 1994c.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Jiři Outrata
    • 1
  • Michal Kočvara
    • 2
  • Jochem Zowe
    • 2
  1. 1.Institute of Information Theory and AutomationCzech Academy of SciencesPragueCzech Republic
  2. 2.Institute of Applied MathematicsUniversity of Erlangen-NurembergErlangenGermany

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