Computing Equilibria and Fixed Points pp 79-100 | Cite as

# Simplicial Homotopy Algorithms

## Abstract

In the previous chapter we introduced the first generation fixed point algorithms. As we see, those algorithms all suffer from computational inefficiency, caused by the following two difficulties: (1) they have to start outside the region of interest; (2) the mesh size of the underlying triangulation is kept constant throughout the whole computing process. To circumvent these impasses, several methods have been developed over the last thirty years. In this chapter we discuss two of them. The first method is that of Merrill [1972] and Kuhn and MacKinnon [1975], called the restart algorithm. This algorithm can start anywhere. When an approximate solution is found, the algorithm can restart from this approximation with a finer triangulation. The second method is that of Eaves [1972] and Eaves and Saigal [1972], called the simplicial homotopy algorithm. This algorithm can start anywhere and the grid size of simplices is automatically refined during the course of the algorithm. To grasp the idea of these methods, let us consider the fixed point problem of a continuous function *f* : *C* ↦ *C*. Recall that the classical homotopy idea in topology goes as follows: Let *f* _{ t }
(*x*) = (1 − *t*)*g*(*x*) + *tf*(*x*) with *t* ∈ [0,1] and *x* ∈ *C*, where *g* : *C* ↦ *C* is a constant function with a known fixed point, say,*x* _{0} ,(i.e.,*g*(*x*) = *x* _{0} for all *x* ∈ *C*.) In order to find a fixed point of *f*, we begin with the constant function *f* _{0}(*x*) = *g*(*x*) and its fixed point *x* _{0}, deform *f* _{ t } as *t* converges to 1 and *f* _{ t } converges to the original function *f*,and follow the path of fixed points *x* _{ t } of *f* _{ t }. Cluster points of the sequence {*x* _{ t } ∣ *t* ∈ Z_{+}} are fixed points of *f*. The reader may have realized that in this framework we need to work in a one dimension higher space, i.e., *C ×* [0,1]. It will become clear that both Merrill’s algorithm and the algorithms of Eaves and (Saigal) are indeed homotopy algorithms but the latter ones are much more natural than the former one. In this chapter we present Merrill’s algorithm first and then the algorithms of Eaves and (Saigal), because we believe the latter algorithms are more elegant and sophisticated than the former one, although Eaves and Merrill discovered their respective algorithms independently and simultaneously.

## Keywords

Simple Path Piecewise Linear Approximation Basis Column Basic Feasible Solution Vector Label## Preview

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