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A vector labeling method for solving discrete zero point and complementarity problems

SIAM JOURNAL ON OPTIMIZATION 2007年第1期18卷 290-308页

作者： Van Der Laan, Gerard Talman, Dolf Yang, Zaifu Vrije Univ Amsterdam Dept Econometr NL-1081 HV Amsterdam Netherlands Vrije Univ Amsterdam Tinbergen Inst NL-1081 HV Amsterdam Netherlands Tilburg Univ Dept Econometr & Operat NL-5000 LE Tilburg Netherlands Tilburg Univ CentER NL-5000 LE Tilburg Netherlands Yokohama Natl Univ Fac Business Adm Yokohama Kanagawa 2408501 Japan

In this paper we establish the existence of a discrete zero point of a function from the n-dimensional integer lattice Z(n) to the n-dimensional Euclidean space R-n under very general conditions with respect to the behavior of the function. The proof is constructive and uses a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps. The algorithm provides an efficient method to find an exact solution. We also discuss how to adapt the algorithm for two related problems, namely, to find a discrete zero point of a function under a general antipodal condition and to find a solution to a discrete nonlinear complementarity problem. In both cases the modified algorithm provides a constructive existence proof, too. We further show that the algorithm for the discrete nonlinear complementarity problem generalizes the well-known Lemke's method to nonlinear environments. An economic application is also presented.

关键词： integer lattice zero point vector labeling rule simplicial algorithm discrete complementarity

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Solving discrete zero point problems

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MATHEMATICAL PROGRAMMING 2006年第1期108卷 127-134页

作者： Van der Laan, G. Talman, D. A. J. J. Yang, Z. Vrije Univ Amsterdam Dept Econometr NL-1081 HV Amsterdam Netherlands Vrije Univ Amsterdam Tinbergen Inst NL-1081 HV Amsterdam Netherlands Tilburg Univ Dept Econometr & Operat Res NL-5000 LE Tilburg Netherlands Tilburg Univ CentER NL-5000 LE Tilburg Netherlands Yokohama Natl Univ Fac Business Adm Yokohama Kanagawa 2408501 Japan

In this paper we present two theorems on the existence of a discrete zero point of a function from the n-dimensional integer lattice Z(n) to the n-dimensional Euclidean space R-n. The theorems differ in their boundary conditions. For both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps.

关键词： discrete zero point labeling rule simplicial algorithm triangulation

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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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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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Variational inequality problems with a continuum of solutions: Existence and computation

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2001年第6期39卷 1852-1873页

作者： Herings, PJJ Talman, D Yang, ZF Univ Maastricht Dept Econ NL-6200 MD Maastricht Netherlands Tilburg Univ Dept Econometr & CentER NL-5000 LE Tilburg Netherlands Yokohama Natl Univ Fac Business Adm Yokohama Kanagawa 2408501 Japan

In this paper three sufficient conditions are provided under each of which an upper semicontinuous point-to-set mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope. Furthermore, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary polytopes. These results follow in a constructive way by designing a new simplicial algorithm. The algorithm operates on a triangulation of the polytope and generates a piecewise linear path of points connecting two distinct faces of the polytope. Each point on the path is an approximate zero point. As the mesh size of the triangulation goes to zero, the path converges to a connected set of zero points linking the two distinct faces. As a consequence, our results generalize Browder's fixed point theorem [ Summa Brasiliensis Mathematicae, 4 (1960), pp. 183-191] and an earlier result by the authors [ Math. Oper. Res., 21 (1996), pp. 675-696] on the n-dimensional unit cube. An application in economics and some numerical examples are also discussed.

关键词： polytope simplicial algorithm continuum of zero points system of nonlinear equations variational inequality economic equilibrium model

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Existence and approximation of robust solutions of variational inequality problems over polytopes

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1999年第2期37卷 333-352页

作者： Van Der Laan, G Talman, D Yang, Z Free Univ Amsterdam Dept Econometrics NL-1081 HV Amsterdam Netherlands Tilburg Univ Dept Econometrics & Ctr NL-5000 LE Tilburg Netherlands Yokohama Natl Univ Fac Business Adm Yokohama Kanagawa 2400067 Japan

We study nonlinear variational inequality problems over polytopes from a viewpoint of stability and propose a new solution concept. Extending an earlier concept proposed by Yang [Z. Yang, SIAM J. Control Optim., 34 (1996), pp. 491-506] on the unit simplex, we will introduce the concept of the robust stationary point, which is a refinement of the concept of the stationary point. Though a stationary point need not be robust, it is shown that every continuous function on a polytope has a robust stationary point. We develop a simplicial algorithm to compute a robust stationary point of a continuous function on a polytope. The algorithm can be briefly stated as follows. Starting with any point in the relative interior of a polytope, the algorithm generates a piecewise linear path which leads to an approximate robust stationary point of any a priori chosen accuracy within a finite number of steps. Moreover, we also discuss several numerical examples and apply the new concept to noncooperative games and economic equilibrium problems.

关键词： variational inequality problem robust stationary point polytopes simplicial algorithm fixed points stability Nash equilibrium Walrasian equilibrium

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A simplicial algorithm for computing robust stationary points of a continuous function on the unit simplex

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1996年第2期34卷 491-506页

作者： Yang, Z TILBURG UNIV CTR ECON RES5000 LE TILBURGNETHERLANDS

A simplicial algorithm is proposed to compute a robust stationary point of a continuous function f from the (n - 1)-dimensional unit simplex S-n-1 into R(n). The concept of robust stationary point is a refinement of the concept of stationary point of f on S-n-1. Starting from an arbitrarily chosen interior point v in S-n-1, the algorithm generates a piecewise linear path of points in S-n-1. This path is followed by alternating linear programming pivot steps and replacement steps in a specific simplicial subdivision of the relative interior of S-n-1 In this way an approximate robust stationary point of any a priori chosen accuracy is reached within a finite number of steps. The algorithm leaves the starting point along one out of n! rays. When the path approaches the boundary of S-n-1, the mesh size of the triangulation along the path automatically goes to zero. This makes the algorithm different from all simplicial restart algorithms and homotopy algorithms known so far. Roughly speaking, the algorithm is a blend of a restart and a homotopy algorithm and maintains the basic properties of both. However, the algorithm does not need an extra dimension as homotopy algorithms do. Some examples are discussed.

关键词： robust stationary point variational inequality simplicial algorithm subdivision piecewise linear approximation stability

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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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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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