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Solving discrete systems of nonlinear equations

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第3期214卷 493-500页

作者： van der Laan, Gerard Talman, Dolf Yang, Zaifu Tilburg Univ CentER Dept Econometr & Operat Res NL-5000 LE Tilburg Netherlands Vrije Univ Amsterdam Dept Econometr NL-1081 HV Amsterdam Netherlands Vrije Univ Amsterdam Tinbergen Inst NL-1081 HV Amsterdam Netherlands Univ York Dept Econ & Related Studies York YO10 5DD N Yorkshire England

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Z(n) of the n-dimensional Euclidean space R-n. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Z(n) and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the 'continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem. (C) 2011 Published by Elsevier By.

关键词： Discrete system of equations Triangulation simplicial algorithm Fixed point Zero point

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A constructive proof of Ky Fan's coincidence theorem

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MATHEMATICAL PROGRAMMING 2009年第2期118卷 317-325页

作者： Talman, A. J. J. Yang, Z. Tilburg Univ Dept Econometr & Operat Res CentER NL-5000 LE Tilburg Netherlands Yokohama Natl Univ Fac Business Adm Yokohama Kanagawa 2408501 Japan

We present a constructive proof for the well-known Ky Fan's coincidence theorem through a simplicial algorithm. In a finite number of steps the algorithm generates a simplex containing an approximate coincidence point. In the limit, when the mesh size converges to zero, the sequence of approximations converges to a coincidence point.

关键词： Fan's coincidence theorem Fixed point simplicial algorithm Upper semi-continuity

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COMPUTATION OF EQUILIBRIA IN AN ECONOMY WITH INCREASING RETURNS TO SCALE TECHNOLOGIES

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MATHEMATICAL PROGRAMMING 1990年第2期49卷 253-261页

作者： KAMIYA, K OSAKA UNIV DEPT ECONTOYONAKAOSAKA 560JAPAN

The purpose of this paper is to show that equilibria in economy with increasing returns to scale technologies are computable.

关键词： COMPUTATION OF EQUILIBRIA INCREASING RETURNS simplicial algorithm

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An adjustment process for nonconvex production economies

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JOURNAL OF MATHEMATICAL ECONOMICS 2006年第1期42卷 1-13页

作者： van den Elzen, A Kremers, H DIW Dept Energy Transportat & Environm EVU D-14195 Berlin Germany

We prove the existence of a path of market conditions, i.e. combinations of market prices and production quantities, that links any arbitrarily chosen market condition with an equilibrium, in a general equilibrium model with possibly nonconvex production technologies based on Villar [Villar, A., 1994. Equilibrium with nonconvex production technologies. Economic Theory 4, 629-638] and Villar [Villar, A., 1999. Equilibrium and Efficiency in Production Economies, second ed., Springer Verlag, Berlin]. This existence theorem holds for any semi-algebraic version of the model and the adjustment of market conditions along the path can be given an economic interpretation as a tatonnement process. Any such path can be approximated arbitrarily close by applying a simplicial algorithm. By restarting this algorithm in a different market condition, we may find more than one equilibrium. (c) 2005 Elsevier B.V. All rights reserved.

关键词： general equilibrium nonconvex production semi-algebraic economy globally convergent adjustment process simplicial algorithm

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The transition from a Dreze equilibrium to a Walrasian equilibrium

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JOURNAL OF MATHEMATICAL ECONOMICS 1998年第3期29卷 303-330页

作者： Herings, JJ van der Laan, G Venniker, R Free Univ Amsterdam Dept Econometr NL-1081 HV Amsterdam Netherlands Free Univ Amsterdam Tinbergen Inst NL-1081 HV Amsterdam Netherlands Tilburg Univ Dept Econometr NL-5000 LE Tilburg Netherlands Tilburg Univ CentER NL-5000 LE Tilburg Netherlands

In this paper a price and quantity adjustment process in continuous time is considered for an economy facing price rigidities. In the short run prices are assumed to be completely fixed and the markets are cleared by quantity adjustments until a fixed price equilibrium is reached where every market is typically characterized by either supply rationing or demand rationing. Using only standard assumptions on the primitive concepts of the economy and a non-degeneracy condition, it is shown that the process indeed converges to a fixed price equilibrium for the initially given prices in the short run. In the long run prices are assumed to move upwards in the case of demand rationing on a market and downwards when supply rationing occurs, while markets are kept in equilibrium by infinitesimal quantity adjustments. Again, under standard assumptions on the primitive concepts of the economy and a non-degeneracy condition, the process is shown to reach a Walrasian equilibrium in the long run. A simplicial algorithm has been developed to make the study of the price and quantity adjustment process possible and the accuracy of this algorithm is discussed. (C) 1998 Elsevier Science S.A.

关键词： exchange economy price rigidities disequilibrium adjustment process simplicial algorithm

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Equilibrium adjustment of disequilibrium prices

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JOURNAL OF MATHEMATICAL ECONOMICS 1997年第1期27卷 53-77页

作者： Herings, JJ vanderLaan, G Talman, D Venniker, R FREE UNIV AMSTERDAM DEPT ECONOMETRNL-1081 HV AMSTERDAMNETHERLANDS FREE UNIV AMSTERDAM TINBERGEN INSTNL-1081 HV AMSTERDAMNETHERLANDS

We consider an exchange economy in which price rigidities are present. An always converging price and quantity adjustment process for such an economy is presented that is based on a discrete algorithmic procedure rather than on more traditional adjustment processes, which are based on difference or differential equations. In the short run, all non-numeraire commodities have a flexible price level with respect to the numeraire commodity but their relative prices are mutually fixed. In the long run, prices are assumed to be completely flexible. The adjustment process starts with a trivial equilibrium with a low enough price level and complete demand rationing on all markets. Along the path followed by the adjustment process, initially all relative prices of the non-numeraire commodities are kept fixed and the price level is increased. Rationing schemes are adjusted to keep markets in equilibrium. In doing so, the process reaches a short-run equilibrium with only demand rationing and no rationing on the numeraire and at least one of the other commodities. In the long run, the process allows for a downward price adjustment of unrationed non-numeraire commodities and eventually reaches a Walrasian equilibrium.

关键词： exchange economy price rigidities disequilibrium simplicial algorithm adjustment process Dreze equilibrium Walrasian equilibrium

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A CONTINUOUS DEFORMATION algorithm FOR VARIATIONAL INEQUALITY PROBLEMS ON POLYTOPES

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MATHEMATICAL PROGRAMMING 1994年第1期64卷 103-122页

作者： DAI, Y YAMAMOTO, Y UNIV TSUKUBA INST SOCIOECON PLANNINGTSUKUBAIBARAKI 305JAPAN KOBE UNIV COMMERCE DEPT MANAGEMENT SCINISHI KUKOBEHYOGO 65112JAPAN

A continuous deformation algorithm is proposed for solving a variational inequality problem on a polytope K. The algorithm embeds the polytope K into K x [0, infinity) and starts by applying a variable dimension algorithm on K x {0} until an approximate solution is found on K x {0}. Then by tracing the path of solutions of a system of equations the algorithm virtually follows a path of approximate solution in K x [0, infinity). When the path in K x [0, infinity) returns to level 0, i.e., K x {0}, the variable dimension algorithm is again used until a new approximate solution is found on K x {0}. The set K x [0, infinity) is triangulated so that the approximate solution on the path improves the accuracy as the level increases. A contrivance for a practical implementation of the algorithm is proposed and tested for some test problems.

关键词： VARIATIONAL INEQUALITY STATIONARY POINT simplicial algorithm VARIABLE DIMENSION algorithm CONTINUOUS DEFORMATION TRIANGULATION

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H ∞ -Norm Computation for Linear System with Parameter Perturbations

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IFAC Proceedings Volumes 1997年第16期30卷 261-266页

作者： Chyi Hwang Jyh-jia Chen Shih-lian Cheng Department of Chemical Engineering National Chung Cheng University Chia-Yi 621 TAIWAN Department of Chemical Engineering National Cheng Rung University Tainan 701 TAIWAN

In this paper, we present a computationally tractable method for constructing the value set G(jR ; Q) of a family of parametric transfer functions G(s; q), where j =, Q is an m-dimensional box in R, and R = [0, ∞). The method is based on characterizing the whole set of generalized principal points in the space R x Q whose image under the mapping G(jw ; q) covers the boundary of the value set G(jR ; Q). By searching the points over the set of generalized principal points, the H-norm for systems with parametric uncertainties can be accurately obtained. Thus, the proposed method makes it feasible to design robust H ∞ controllers for uncertain linear systems having a general structure of parameter perturbations, including uncertain time delays. To illustrate the effectiveness of the proposed method, an illustrative example is provided.

关键词： Robust Nyquist plot H-norm uncertain time delays simplicial algorithm linear systems parametric uncertainty

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A New Constructive Proof to the Existence of an Integer Zero Point of a Mapping with the Direction Preserving Property

A New Constructive Proof to the Existence of an Integer Zero...

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第八届运筹学及其应用国际研讨会

作者： Chuangyin Dang Guixian Zhong Dept.of Manufacturing Engineering & Engineering Management City University of Hong Kong

Let f:Z~n→R~n be a mapping satisfying the direction preserving property that f（x） > 0 implies f（y）≥0 for any integer points x and y with ||x-y||∞≤*** assume that there is an integer point x with c≤x≤d satisfying that （?）(x-x)f（x） >0 for any integer point x with f（x）≠0 on the boundary of H = {x∈R~n|c-e≤x≤d + e},where c and d are two finite integer points with c≤d and e =（1,1,…,1）∈R~*** assumption is implied by one of two different conditions for the existence of an integer zero point of the mapping in van der Laan et al.（2004）.Under the assumption,there is an integer point x*∈H such that f{x*) = 0.A constructive proof of the existence is derived from an application of the well-known （n +1）-ray algorithm for computing a fixed *** existence result has applications in general equilibrium models with indivisible commodities.

关键词： Integer Zero Point Direction Preserving simplicial algorithm Triangulation Existence

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基于三角域的B样条建模：理论与实践

基于三角域的B样条建模：理论与实践

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作者：杨君浙江大学

学位级别：硕士

在过去的35年中,多元样条函数在计算机辅助几何设计(Computer Aided Geometric Design)和数值计算(Numerical Computation)领域中被许多学者进行了大量的研究。至今为止,许多关于多元样条的文章和专著作已经发表和出版。多元样条已被广... 详细信息

在过去的35年中,多元样条函数在计算机辅助几何设计(Computer Aided Geometric Design)和数值计算(Numerical Computation)领域中被许多学者进行了大量的研究。至今为止,许多关于多元样条的文章和专著作已经发表和出版。多元样条已被广泛应用于许多现代工业和学术领域中,其中包括计算机图形学和视觉、数字图像处理、计算机动画、计算机辅助工业制造、计算机辅助几何建模、逆向工程、逼近论、计算几何等。计算机辅助几何设计已经从应用数学的一个分支发展成为一门独立的交叉学科。在计算机辅助几何设计领域中,许多种参数曲线和曲面已经有了完整的理论。因为单纯形是任意维实仿射空间中最简单的多面体,因此许多学者对于基于三角剖分域多元参数曲面非常感兴趣。犹于任意高维实仿射空间的多面体都可以被分解成一个由相领单纯形构成的集合,因此有了上面的曲面,那么我们可以定义任意高维仿射空间多面体参数域上的曲面。不少基于三角剖分的曲面已经被提出了,但是最基本的多元B样条-纯形样条,并没有被广泛应用。这主要根源于实现它时的困难。三角B样条(Triangular B-Splines)曲面是以规范的单纯形样条做为基函数的三角剖分上的曲面,因此它们可以定义在定义域空间中任意拓扑定义域上。我们尝试来填充三角B样条曲面和理论和实现上的差距。我们已经实现了一个二元三次三角B样条曲面平台,实践已经证明该平台上的效率足以使用三角B样条曲面在实际中进行建模。在我们的平台中,三角B样条的几何特性被赋于了特别的重视。因为这样的设计会使得建立更高维和更高阶的三角B样条可以从现在的平台上容易的扩展。本文中,三角B样条曲面是基于几何进行介绍的,这并不同于以前仅仅做三角B样条曲面的实现。一些我们在实践中发现的有用的命题和性质在文中被进行了系统的总结。这些命题和性质都是基于对三角B样条的理论背景进行深入研究并且结合实践而发现的。同时,我们对于blossoming在文中也进行了深入的讨论。总而言之,我们使用了从理论到实践的方法来设计我们的平台,而且实践证明平台的效率足以让三角B样条被应用于实际建模中去。

关键词：三角B样条曲面 B-patch 多元B样条单纯形样条 blossoming simplicial algorithm

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