First Algorithms for Approximating Fixed Points on the Unit Simplex
In this chapter we consider the problem of computing fixed points of a continuous function f from the unit simplex S n into itself. As we see in Chapter 1, Brouwer theorem guarantees the existence of a fixed point on S n but does not offer any insight about the location of such fixed points. Nevertheless it is clear to us that in order to approximate a fixed point, one possible way is to design a systematic procedure for finding completely labelled simplices. Such procedure did not exist until 1967. In his pioneering work Scarf [1967b] developed the first such procedure. Starting with a specific primitive set, this procedure generates a path of adjacent primitive sets and terminates within a finite number of steps with a completely labelled primitive set which yields a good approximation of a fixed point. (Completely labelled primitive sets are analogue of completely labelled simplices.) To prove the convergence of the procedure, Scarf utilized an argument of Lemke and Howson  and Lemke  which is a purely combinatorial argument and does not rely on any monotonicity property. Later Kuhn [1968,1969] proposed two alternative procedures by using simplices and triangulations instead of primitive sets. These very first fixed point methods are the central topic of this chapter.
KeywordsNected Component Unit Simplex Fixed Point Algorithm Pivot Rule Label Rule
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