The Computation of Fixed Points and Applications (Lecture Notes in Economics and Mathematical Systems, 124),Used

The Computation of Fixed Points and Applications (Lecture Notes in Economics and Mathematical Systems, 124),Used

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Fixedpoint algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundaryvalue 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 fixedpoint 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 fixedpoint 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 fixedpoint 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 fixedpoint 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 fixedpoint algorithms to find many solutions to boundaryvalue problems.

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