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A solution condition for complementarity problems: With an application to spatial price equilibrium. (English) Zbl 0545.90094

The paper gives a sufficient condition for the existence of solutions to the nonlinear complementarity problem (find a nonnegative vector x whose image by a given continuous function is both nonnegative, and orthogonal to x). This existence condition is shown to be weaker than the coerciveness condition.
The result is applied to the following problem in spatial equilibrium: Given a commodity sold on a number of markets linked by a common transportation network, find a vector of prices and a vector of flows such that trading between each pair of markets is efficient, and such that supply equals demand in each market. Previous work proved the existence of such a spatial price equilibrium under the assumption of bounded demand. As shown by the author, the new existence condition for the complementarity problem makes that assumption unnecessary.
Reviewer: P.J.Deschamps

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C90 Applications of mathematical programming
91B50 General equilibrium theory
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
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References:

[1] H.Z. Aashtiani and T.L. Magnanti, Equilibria on a congested transportation network, SIAM J. Algebriac Discrete Methods; H.Z. Aashtiani and T.L. Magnanti, Equilibria on a congested transportation network, SIAM J. Algebriac Discrete Methods · Zbl 0501.90033
[2] 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, J. Regional Sci.; 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, J. Regional Sci.
[3] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and their Applications (1980), Academic: Academic New York · Zbl 0457.35001
[4] Lembke, C. E., A survey of complementarity theory, (Cottle, R. W.; Ginanessi, F.; Lions, J-L., Variational Inequalities and Complementarity Problems (1980), Wiley: Wiley New York)
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