The Computation of Fixed Points and Applications

ISBN-13: 9783540076858 / Angielski / Miękka / 1976 / 132 str.

M. J. Todd

Opis

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.

Spis treści

I Brouwer’s Theorem.- II Some Applications of Brouwer’s Theorem.- III Triangulations.- IV Algorithms to Find Completely-Labelled Simplices.- V Extensions of Brouwer’s Theorem.- VI Applications of Kakutani’s Theorem and its Extensions.- VII Eaves’ First Algorithm.- VIII Merrill’s Algorithm.- IX Homotopy Algorithms.- X Triangulations with Continuous Refinement of Grid Size.- XI Measures of Efficiency for Triangulations.- References.