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Refinement and Stability of Stationary Points

  • Zaifu Yang
Part of the Theory and Decision Library book series (TDLC, volume 21)

Abstract

The purpose of this chapter is twofold. On the one hand, a new solution to the stationary point problem or the variational inequality problem on the unit simplex will be introduced. This new solution concept is called a robust stationary point. It is a refinement of the concept of stationary point and was essentially motivated from economic equilibrium problems, noncooperative games, biological and engineering problems. We recommend the interested reader to Arrow and Hurwicz [1958], Arrow, Block and Hurwicz [1959], Wu and Jiang [1962], Selten [1975], Myerson [1978], Kreps and Wilson [1982], and van Damme [1987] for the various motivations. Mathematically speaking, a continuous function from the unit simplex S n into ℝ n may have multiple stationary points and some of them are undesirable from a point of view of stability. So it is important to eliminate those undesirable stationary points. One way of achieving this goal is to refine the concept of a stationary point. It will be shown that every continuous function on the unit simplex has at least one stationary point, although a stationary point need not be robust. It will also be shown that when applying this refined concept to game-theoretic or economic equilibrium problems, it is very meaningful and intuitive. On the other hand, we shall spend a fairly large portion of time on the computation of robust stationary points. To do so, a new simplicial algorithm will be developed. This algorithm is called an adaptive simplicial algorithm.

Keywords

Stationary Point Variational Inequality Problem Piecewise Linear Approximation Pivot Step Unit Simplex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Zaifu Yang
    • 1
  1. 1.Yokohama National UniversityJapan

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