Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimizations

Simplicial methods were originated by Scarf for approximating fixed points of continuous mappings. They have many applications in economics and science. With regard to its geometric structure, a simplicial method can be classified as either a variable dimension simplicial method or a homotopy simplicial method. A variable dimension simplicial method works directly on the interested space, whereas a simplicial homotopy method needs to introduce an extra dimension. It is well known that integer programming is equivalent to determining whether there is an integer point in a polytope. Simplicial methods were extended to computing an integer point in a polytope. There is a significant difference between simplicial methods for approximating fixed points and simplicial methods for integer programming, though they both have the same foundation. Two most important components of simplicial methods are labeling rules and triangulations. Efficiency of simplicial methods depends critically on the underlying triangulations. Three simplest triangulations of Rⁿ are the K_i-triangulation, the J₁-triangulation, and the D₁-triangulation. This chapter presents a brief introduction to these developments of simplicial methods.