Ebook: Simplicial Algorithms on the Simplotope

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.

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This monograph deals with simplicial algorithms on the unit simplex and on the simplotope. Several new triangulations are introduced underlying the simplicial algorithms. The V-triangulation underlies a number of new simplicial algorithms and also underlies a continuous deformation algorithm on the simplotope. The monograph extensively discusses these algorithms and gives computational comparisons. The examples include exchange economies, quadratic programming, non-cooperative N-person games, and economies with a block diagonal supply-demand pattern. In the economic examples, the paths induced by the algorithms can be interpreted as price adjustment processes. These paths have the attractive feature that they always converge to an optimal solution.

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This monograph deals with simplicial algorithms on the unit simplex and on the simplotope. Several new triangulations are introduced underlying the simplicial algorithms. The V-triangulation underlies a number of new simplicial algorithms and also underlies a continuous deformation algorithm on the simplotope. The monograph extensively discusses these algorithms and gives computational comparisons. The examples include exchange economies, quadratic programming, non-cooperative N-person games, and economies with a block diagonal supply-demand pattern. In the economic examples, the paths induced by the algorithms can be interpreted as price adjustment processes. These paths have the attractive feature that they always converge to an optimal solution.
Content:
Front Matter....Pages N2-VIII
Front Matter....Pages 1-1
Introduction....Pages 3-13
Definitions and Existence Theorems....Pages 15-44
Triangulations of Sn and S....Pages 45-65
Front Matter....Pages 67-67
An Introduction to Simplicial Algorithms on the Unit Simplex....Pages 69-93
The (2n+1 ?2)-Ray Algorithm....Pages 95-110
The 2-Ray Algorithm....Pages 111-126
Comparisons and Computational Results....Pages 127-133
Front Matter....Pages 135-135
An Introduction to Simplicial Algorithms on the Simplotope....Pages 137-158
The Product-Ray Algorithm....Pages 159-172
The Exponent-Ray Algorithm....Pages 173-195
Comparisons and Computational Results....Pages 197-203
Front Matter....Pages 205-205
The Continuous Deformation Algorithm on the Simplotope....Pages 207-254
Back Matter....Pages 255-264

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This monograph deals with simplicial algorithms on the unit simplex and on the simplotope. Several new triangulations are introduced underlying the simplicial algorithms. The V-triangulation underlies a number of new simplicial algorithms and also underlies a continuous deformation algorithm on the simplotope. The monograph extensively discusses these algorithms and gives computational comparisons. The examples include exchange economies, quadratic programming, non-cooperative N-person games, and economies with a block diagonal supply-demand pattern. In the economic examples, the paths induced by the algorithms can be interpreted as price adjustment processes. These paths have the attractive feature that they always converge to an optimal solution.
Content:
Front Matter....Pages N2-VIII
Front Matter....Pages 1-1
Introduction....Pages 3-13
Definitions and Existence Theorems....Pages 15-44
Triangulations of Sn and S....Pages 45-65
Front Matter....Pages 67-67
An Introduction to Simplicial Algorithms on the Unit Simplex....Pages 69-93
The (2n+1 ?2)-Ray Algorithm....Pages 95-110
The 2-Ray Algorithm....Pages 111-126
Comparisons and Computational Results....Pages 127-133
Front Matter....Pages 135-135
An Introduction to Simplicial Algorithms on the Simplotope....Pages 137-158
The Product-Ray Algorithm....Pages 159-172
The Exponent-Ray Algorithm....Pages 173-195
Comparisons and Computational Results....Pages 197-203
Front Matter....Pages 205-205
The Continuous Deformation Algorithm on the Simplotope....Pages 207-254
Back Matter....Pages 255-264
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