Triangulations are used in simplicial algorithms to find the fixed points of continuous functions or upper semicontinuous mappings;applications arise from economics and optimization. The performance of simplicial algo...
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Triangulations are used in simplicial algorithms to find the fixed points of continuous functions or upper semicontinuous mappings;applications arise from economics and optimization. The performance of simplicial algorithms is very sensitive to the triangulation used. Using a facetal description, Dang's D1 triangulation is modified to obtain a more efficient triangulation of the unit hypercube in R(n), and then, by means of translations and reflections, we derive a new triangulation, D1', of R(n). It is shown that D1' uses fewer simplices (asymptotically 30 percent fewer) than D1 while achieving comparable scores for other performance measures such as the diameter and the surface density. The results of Haiman's recursive method for getting asymptotically better triangulations from D1, D1' and other triangulations are also compared
This paper discusses a new method for the stability analysis of a nonlinear system using simplicial algorithms. The system analyzed is a unstable first order plant, driven by a Bang-Bang actuator, stabilized by a nonl...
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This paper discusses a new method for the stability analysis of a nonlinear system using simplicial algorithms. The system analyzed is a unstable first order plant, driven by a Bang-Bang actuator, stabilized by a nonlinear adaptive perturbation filter. A describing function model of the actuator is used in this analysis. This system is implemented on a digital computer, and in analog circuit form, to demonstrate the practicality of the method.
This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and whi...
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We examine the efficiency of PL path following algorithms in followingF T -1 (0), whereF T is the PL approximation, induced by the simplicial triangulationT, to a mapf:? n →? n-1. In particular, we consider the prob...
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We examine the efficiency of PL path following algorithms in followingF T-1 (0), whereF T is the PL approximation, induced by the simplicial triangulationT, to a mapf:? n →? n-1. In particular, we consider the problem of determining an upper bound on the expected number of pivots made per unit length off ?1(0) that is approximated. We show that if the sizes of the simplices ofT are “sufficiently small”, where “sufficiently small” is an explicitly given quantity dependent on measurements of how “nice”f is, then the average directional density ofT, as introduced by Todd, really does give a good approximation to the expected number of pivots made, confirming what researchers have believed on intuitive grounds for a decade. Because what constitutes “sufficiently small” is a precisely given quantity, i.e., non-asymptotic, we are able to provide some rigorous justification for the claim that the expected number of pivots grows only polynomially inn, the number of variables.
In this paper, we consider the limiting paths of simplicial algorithms for finding a zero point. By rewriting the zero-point problem as a problem of finding a stationary point, the problem can be solved by generating ...
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In this paper, we consider the limiting paths of simplicial algorithms for finding a zero point. By rewriting the zero-point problem as a problem of finding a stationary point, the problem can be solved by generating a path of stationary points of the function restricted to an expanding convex, compact set. The limiting path of a simplicial algorithm to find a zero point is obtained by choosing this set in an appropriate way. Almost all simplicial algorithms fit in this framework. Using this framework, it can be shown very easily that Merrill's condition is sufficient for convergence of the algorithms.
Recently Zangwill and Garcia introduced a general formulation of equilibrium problems. To prove the existence of an equilibrium they discussed a path following procedure. In this note we consider the application to th...
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Recently Zangwill and Garcia introduced a general formulation of equilibrium problems. To prove the existence of an equilibrium they discussed a path following procedure. In this note we consider the application to the exchange economy problem. An economic equilibrium may be found by applying a simplicial variable dimension algorithm developed by Van der Laan and *** will show that when an approriate triagulation and labelling rule is taken the limiting path of this algorithm coincides with the adjustment process induced by the procedure of Zangwill and Garcia. This process has a plausible economic interpretation and is an attractive alternative for the well-known tâtonnement process.
Call a subset of R[supn] complete if the origin is in its convex hull. We are concerned with the construction of large finite sets of points on the unit sphere whose complete subsets are, in some sense, easy to descri...
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Call a subset of R[supn] complete if the origin is in its convex hull. We are concerned with the construction of large finite sets of points on the unit sphere whose complete subsets are, in some sense, easy to describe. Our motivation comes from simplicial algorithms for approximating zeroes of functions. We construct two classes of such subsets of the sphere, both of which generalize a subset that corresponds to usual integer labelling. [ABSTRACT FROM AUTHOR]
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