The Computation Of Fixed Points And Applications

The Computation of Fixed Points and Applications

Fixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Since Scarf's pioneering work [56,57] on obtaining approximate fixed points of continuous mappings, a great deal of research has been done in extending the applicability and improving the efficiency of fixed-point methods. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods. However, the algorithms described by Scarf have been super~eded by the more sophisticated restart and homotopy techniques of Merrill [~8,~9] and Eaves and Saigal [1~,16]. To understand the more efficient algorithms one must become familiar with the notions of triangulation and simplicial approxi- tion, whereas Scarf stresses the concept of primitive set. These notes are intended to introduce to a wider audience the most recent fixed-point methods and their applications. Our approach is therefore via triangu- tions. For this reason, Scarf is cited less in this manuscript than his contri- tions would otherwise warrant. We have also confined our treatment of applications to the computation of economic equilibria and the solution of optimization problems. Hansen and Koopmans [28] apply fixed-point methods to the computation of an invariant optimal capital stock in an economic growth model. Applications to game theory are discussed in Scarf [56,58], Shapley [59], and Garcia, Lemke and Luethi [24]. Allgower [1] and Jeppson [31] use fixed-point algorithms to find many solutions to boundary-value problems.

The computation of fixed points and applications

Analysis and Computation of Fixed Points

Analysis and Computation of Fixed Points contains the proceedings of a Symposium on Analysis and Computation of Fixed Points, held at the University of Wisconsin-Madison on May 7-8, 1979. The papers focus on the analysis and computation of fixed points and cover topics ranging from paths generated by fixed point algorithms to strongly stable stationary solutions in nonlinear programs. A simple reliable numerical algorithm for following homotopy paths is also presented. Comprised of nine chapters, this book begins by describing the techniques of numerical linear algebra that possess attractive stability properties and exploit sparsity, and their application to the linear systems that arise in algorithms that solve equations by constructing piecewise-linear homotopies. The reader is then introduced to two triangulations for homotopy fixed point algorithms with an arbitrary grid refinement, followed by a discussion on some generic properties of paths generated by fixed point algorithms. Subsequent chapters deal with topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems; general equilibrium analysis of taxation policy; and solving urban general equilibrium models by fixed point methods. The book concludes with an evaluation of economic equilibrium under deformation of the economy. This monograph should be of interest to students and specialists in the field of mathematics.