# Simplicial Algorithms on the Simplotope A second way to follow the curve is by using a simplicial homotopy algorithm. This homotopy method works as follows. The space is subdivided into simplices and the homotopy function is linearized on each of these simplices.

Author: Timothy M. Doup

Publisher: Springer Science & Business Media

ISBN: 9783642466519

Page: 262

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1.1. Introduction Solving systems of nonlinear equations has since long been of great interest to researchers in the field of economics, mathematics, en gineering, and many other professions. Many problems such as finding an equilibrium, a zero point, or a fixed point, can be formulated as the problem of finding a solution to a system of nonlinear equations. There are many methods to solve the nonlinear system such as Newton's method, the homotopy method, and the simplicial method. In this monograph we mainly consider the simplicial method. Traditionally, the zero point and fixed point problem have been solved by iterative methods such as Newton's method and modifications thereof. Among the difficulties which may cause an iterative method to perform inefficiently or even fail are: the lack of good starting points, slow convergence, and the lack of smoothness of the underlying function. These difficulties have been partly overcome by the introduction of homo topy methods.

# Triangulations and Simplicial Methods Simplicial Algorithm on the Simplotope. Lecture Notes in Economics and Mathematical Systems 318, Springer-Verlag, Berlin. . Doup, T.M., Elzen, A.H. van den, and Talman, A.J.J.(1987). Simplicial algorithms for solving the nonlinear ...

Author: Chuangyin Dang

Publisher: Springer Science & Business Media

ISBN: 9783642487750

Page: 196

View: 834

As a new type of technique, simplicial methods have yielded extremely important contributions toward solutions of a system of nonlinear equations. Theoretical investigations and numerical tests have shown that the performance of simplicial methods depends critically on the triangulations underlying them. This monograph describes some recent developments in triangulations and simplicial methods. It includes the D1-triangulation and its applications to simplicial methods. As a result, efficiency of simplicial methods has been improved significantly. Thus more effective simplicial methods have been developed.

# Adjustment Processes for Exchange Economies and Noncooperative Games Debreu, G. . , "Excess demand functions", Journal of Mathematical Economics 1, 15-23. Doup, T.M. , Simplicial Algorithms on the Simplotope, Lecture Notes in Economics and Mathematical Systems 318, Springer-Verlag, Berlin.

Author: Antoon van den Elzen

Publisher: Springer Science & Business Media

ISBN: 9783642481338

Page: 146

View: 491

This monograph is a treatise on adjustment processes. We consider price adjustment processes in exchange economies and strategy adjustment processes in noncooperative games. In the most simple version of an exchange economy, i.e. a pure exchange economy, there exist markets on which prices are determined by the demand and supply created by a finite number of consumers willing to exchange their initial endowments in order to maximize their utilities. An equilibrium situation is attained if, for some price vector, demand equals supply in all markets. Starting from a situation not being an equi librium an adjustment process reaches an equilibrium via adaptations of prices. The advantage of the adjustment processes we will present in this monograph is that they exist and converge under far weaker assumptions than existing processes. The second subject concerns the problem of finding Nash equilibria in noncooperative games. A Nash equilibrium is a situation from which no player can improve his position by unilaterally changing his strategy. We present a new algorithm for finding such equilibria. The sequence of stra tegy vectors generated by the algorithm can be interpreted as the path followed by a strategy adjustment process.

# Introduction to Numerical Continuation Methods Doup, T. M. (1988): Simplicial algorithms on the simplotope. ... Doup, T. M. & van der Elzen, A. H. & Talman, A. J. J. (1987): Simplicial algorithm for Solving the nonlinear complementarity problem on the simplotope.

Author: Eugene L. Allgower

Publisher: SIAM

ISBN: 9780898715446

Category: Mathematics

Page: 388

View: 902

Numerical continuation methods have provided important contributions toward the numerical solution of nonlinear systems of equations for many years. The methods may be used not only to compute solutions, which might otherwise be hard to obtain, but also to gain insight into qualitative properties of the solutions. Introduction to Numerical Continuation Methods, originally published in 1979, was the first book to provide easy access to the numerical aspects of predictor corrector continuation and piecewise linear continuation methods. Not only do these seemingly distinct methods share many common features and general principles, they can be numerically implemented in similar ways. Introduction to Numerical Continuation Methods also features the piecewise linear approximation of implicitly defined surfaces, the algorithms of which are frequently used in computer graphics, mesh generation, and the evaluation of surface integrals.
Categories: Mathematics

# Numerical Continuation Methods Doup, T. M. (1988): Simplicial algorithms on the simplotope. ... Doup, T. M. & van der Elzen, A. H. & Talman, A. J. J. (1987): Simplicial algorithm for solving the nonlinear complementarity problem on the simplotope.

Author: Eugene L. Allgower

Publisher: Springer Science & Business Media

ISBN: 9783642612572

Category: Mathematics

Page: 388

View: 543

Over the past fifteen years two new techniques have yielded extremely important contributions toward the numerical solution of nonlinear systems of equations. This book provides an introduction to and an up-to-date survey of numerical continuation methods (tracing of implicitly defined curves) of both predictor-corrector and piecewise-linear types. It presents and analyzes implementations aimed at applications to the computation of zero points, fixed points, nonlinear eigenvalue problems, bifurcation and turning points, and economic equilibria. Many algorithms are presented in a pseudo code format. An appendix supplies five sample FORTRAN programs with numerical examples, which readers can adapt to fit their purposes, and a description of the program package SCOUT for analyzing nonlinear problems via piecewise-linear methods. An extensive up-to-date bibliography spanning 46 pages is included. The material in this book has been presented to students of mathematics, engineering and sciences with great success, and will also serve as a valuable tool for researchers in the field.
Categories: Mathematics

# Static and Dynamic Aspects of General Disequilibrium Theory Doup, T.M. (1988), Simplicial Algorithms on the Simplotope, Lecture Notes in Economics and Mathematical Systems, 318, Springer-Verlag, Berlin. Doup, T.M., A.H. VAN DEN ELZEN, AND A.J.J. TALMAN (1987), “Simplicial Algorithms for Solving ...

Author: P. Jean-Jacques Herings

Publisher: Springer Science & Business Media

ISBN: 9781461562511

Page: 508

View: 550

Mathematical economics uses mathematical tools and reasoning to describe and explain economic reality. At the core of mathematical economics is general equilibrium theory. Static and Dynamic Aspects of General Disequilibrium Theory describes and analyses various general equilibrium models, treating theory from an axiomatic point of view, which may lead to a deeper understanding of problems, may help to avoid incorrect reasoning, and may improve communication within the economic science. This volume consists of four parts, each of which is self-contained. Part I deals with the mathematical and economic preliminaries. Part II considers the static aspects of disequilibrium theory. Part III determines price rigidities endogenously. Finally, Part IV deals with dynamic aspects of disequilibrium theory.

# Algebraic and Differential Topology of Robust Stability T. Doup, Simplicial Algorithms on the Simplotope, Springer-Verlag, New York, 1988, page 12. SUMMARY This chapter is in the same spirit as the computational geometry algorithm for searching the crossover, as developed in Chapter 6.

Author: Edmond A. Jonckheere

Publisher: Oxford University Press

ISBN: 019535768X

Category: Mathematics

Page: 624

View: 293

In this book, two seemingly unrelated fields -- algebraic topology and robust control -- are brought together. The book develops algebraic/differential topology from an application-oriented point of view. The book takes the reader on a path starting from a well-motivated robust stability problem, showing the relevance of the simplicial approximation theorem and how it can be efficiently implemented using computational geometry. The simplicial approximation theorem serves as a primer to more serious topological issues such as the obstruction to extending the Nyquist map, K-theory of robust stabilization, and eventually the differential topology of the Nyquist map, culminating in the explanation of the lack of continuity of the stability margin relative to rounding errors. The book is suitable for graduate students in engineering and/or applied mathematics, academic researchers and governmental laboratories.
Categories: Mathematics

# Computing Equilibria and Fixed Points T.M. Doup , Simplicial Algorithms on the Simplotope, Lecture Notes in Economics and Mathematical Systems 318, Springer-Verlag, Berlin. T.M. Doup, G. van der Laan and A.J.J. Talman , “The (2” – 2)-ray algorithm: a new ...

Author: Zaifu Yang

Publisher: Springer Science & Business Media

ISBN: 9781475748390

Page: 344

View: 388

Computing Equilibria and Fixed Points is devoted to the computation of equilibria, fixed points and stationary points. This volume is written with three goals in mind: (i) To give a comprehensive introduction to fixed point methods and to the definition and construction of Gröbner bases; (ii) To discuss several interesting applications of these methods in the fields of general equilibrium theory, game theory, mathematical programming, algebra and symbolic computation; (iii) To introduce several advanced fixed point and stationary point theorems. These methods and topics should be of interest not only to economists and game theorists concerned with the computation and existence of equilibrium outcomes in economic models and cooperative and non-cooperative games, but also to applied mathematicians, computer scientists and engineers dealing with models of highly nonlinear systems of equations (or polynomial equations).

# Recent Advances in Nonsmooth Optimization Therefore , we conclude the algorithm terminates at a complete simplex in { 1 } x R " within a finite number of ... [ 3 ] T. M. Doup , Simplicial Algorithms on The Simplotope , Lecture Notes on Economics and Mathematical Systems 318 ...

Author: Dingzhu Du

Publisher: World Scientific

ISBN: 9810222653

Category: Mathematics

Page: 472

View: 282

Nonsmooth optimization covers the minimization or maximization of functions which do not have the differentiability properties required by classical methods. The field of nonsmooth optimization is significant, not only because of the existence of nondifferentiable functions arising directly in applications, but also because several important methods for solving difficult smooth problems lead directly to the need to solve nonsmooth problems, which are either smaller in dimension or simpler in structure.This book contains twenty five papers written by forty six authors from twenty countries in five continents. It includes papers on theory, algorithms and applications for problems with first-order nondifferentiability (the usual sense of nonsmooth optimization) second-order nondifferentiability, nonsmooth equations, nonsmooth variational inequalities and other problems related to nonsmooth optimization.
Categories: Mathematics

# Automatic Control in Aerospace 1994 Aerospace Control 94 Simplicial Algorithms on the Simplotope, Lecture Notes in Economics and Mathematical Systems — Number 318, Springer-Verlag, New York, New York. Jonckheere, E. A., and J. R. Bar—on (1991). Algebraic Topology in Robust Control, ...

Author: D. Schaechter

Publisher: Elsevier

ISBN: 9781483296920

Category: Technology & Engineering

Page: 438

View: 988

An important, successful area for control systems development is that of state-of-the-art aeronautical and space related technologies. Leading researchers and practitioners within this field have been given the opportunity to exchange ideas and discuss results at the IFAC symposia on automatic control in aerospace. The key research papers presented at the latest in the series have been put together in this publication to provide a detailed assessment of present and future developments of these control system technologies.
Categories: Technology & Engineering