This book provides the first general account of the development of simplicial algorithms. These include the ubiquitous simplex method of linear programming widely used in industrial optimization and strategic decision making and methods important in data analysis, such as problems involving very large data sets. The theoretical development is based on a new way of representing the underlying geometry of polyhedra functions (functions whose graphs are made up of plane faces), and is capable of resolving problems that occur when combinatorially large numbers of faces intersect at each vertex.