Simplicial Algorithms on the Simplotope

ISBN-13: 9783540502333 / Angielski / Miękka / 1988 / 262 str.

Timothy M. Doup

Opis

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."

Spis treści

I Introduction and Definitions.- 1. Introduction.- 1.1. Introduction.- 1.2. Historical perspective.- 1.3. Outline of the monograph.- 2. Definitions and Existence Theorems.- 2.1. Introduction.- 2.2. Basic concepts and notations.- 2.3. Existence theorems.- 2.4. Labelling functions and accuracy.- 2.5. Pure exchange economies.- 2.6. Quadratically constrained quadratic programming.- 2.7. Economies with a block diagonal supply-demand pattern.- 2.8. Noncooperative N-person games.- 3. Triangulations of Sn and S.- 3.1. Introduction.- 3.2. The Q-triangulation of Sn and S.- 3.3. The Q’-triangulation of S.- 3.4. The V-triangulation of Sn.- 3.5. The V’- and the V-triangulation of S.- 3.6. Variants of the V-triangulation.- II Algorithms on the Unit Simple.- 4. An introduction to Simplicial Algorithms on the Unit Simplex.- 4.1. Introduction.- 4.2. The variable dimension restart algorithm on Sn for proper integer labelling rules.- 4.3. Variable dimension restart algorithms on Sn for arbitrary integer labelling rules.- 4.4. Variable dimension restart algorithms on Sn for vector labelling.- 4.5. A path following interpretation of the variable dimension restart algorithm for the V-triangulation.- 5. The (2n+1-2)-Ray Algorithm.- 5.1. Introduction.- 5.2. The path of the algorithm.- 5.3. The subdivision of Sn.- 5.4. The steps of the algorithm.- 6. The 2-Ray Algorithm.- 6.1. Introduction.- 6.2. The path of the algorithm.- 6.3. The subdivision of Sn.- 6.4. The steps of the algorithm.- 7. Comparisons and Computational Results.- 7.1. Introduction.- 7.2. A comparison of the variable dimension restart algorithms on Sn.- 7.3. Computational results.- III Algorithms on the Simplotope.- 8. An Introduction to Simplicial Algorithms on the Simplotope.- 8.1. Introduction.- 8.2. The sum-ray algorithm on S for proper integer labelling rules.- 8.3. Variable dimension restart algorithms on S for arbitrary integer labelling rules.- 8.4. The sum-ray algorithm on S for vector labelling.- 8.5. A path following interpretation of the sum-ray algorithm for the V’ -triangulation.- 9. The Product-Ray Algorithm.- 9.1. Introduction.- 9.2. The path of the algorithm.- 9.3. The subdivision of S.- 9.4. The steps of the algorithm.- 10. The Exponent-Ray Algorithm.- 10.1. Introduction.- 10.2. The path of the algorithm.- 10.3. The subdivision of S.- 10.4. The steps of the algorithm.- 11. Comparisons and Computational Results.- 11.1. Introduction.- 11.2. A comparison of the variable dimension restart algorithms on S.- 11.3. Computational results.- IV Continuous Deformation on the Simplotope.- 12. The Continuous Deformation Algorithm on the Simplotope.- 12.1. Introduction.- 12.2. The path of the algorithm.- 12.3. Triangulation of S x [1,?).- 12.4. Triangulation of the boundary of S x [1, ?).- 12.5. The steps of the continuous deformation algorithm on S.- References.