# 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
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