Variable Dimension Restart Algorithms

Abstract

In the previous chapter we discussed two efficient methods that are the restart method of Merrill and the homotopy method of Eaves (and Saigal). In both methods we need to imbed an n-dimensional problem into (n + 1)-dimensional space. Moreover, a path of (n + 1)-simplices is generated. In this chapter we will introduce a more efficient algorithm which was developed by van der Laan and Talman [1979]. This algorithm is called the variable dimension restart algorithm. It does not need an extra dimension. More precisely, this algorithm starts with any point in the region of interest (say, Sⁿ⁺¹ or ℞ⁿ), generates a path of adjacent t-simplices of varying dimension and terminates with a completely labelled simplex yielding an approximation of a fixed point. Notice that t can vary between 0 and n. Because the starting point can be arbitrarily chosen in the region of interest, this method is called the variable dimension restart algorithm. So when the accuracy of the current approximate solution is not satisfactory, the algorithm can be restarted at this approximate solution with a finer triangulation in the hope that within a small number of steps a better approximate solution is found. In particular, when the algorithm starts with the point e(1) of the unit simplex, it operates in the same way as Kuhn’s variable dimension algorithm given in Chapter 3 does.