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A NEW variable dimension algorithm FOR THE FIXED-POINT PROBLEM

MATHEMATICAL PROGRAMMING 1983年第3期25卷 329-342页

作者： YAMAMOTO, Y Institute of Socio-Economic Planning The University of Tsukuba Ibaraki Japan

We propose a new variable dimension algorithm for solving a system ofn equations inn variables. The algorithm solves a sequence of subproblems consisting of the first several equations, the dimension of subproblems do... 详细信息

关键词： Fixed Point Piecewise Linear Equations System of Nonlinear Equations variable dimension algorithm

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variable dimension algorithmS - BASIC THEORY, INTERPRETATIONS AND EXTENSIONS OF SOME EXISTING METHODS

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MATHEMATICAL PROGRAMMING 1982年第2期24卷 177-215页

作者： KOJIMA, M YAMAMOTO, Y UNIV TSUKUBA INST SOCIO ECON PLANNINGSAKURAIBARAKI 305JAPAN

In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept ‘primal—dual pair of subdivided manifolds’ and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed.

关键词： Fixed Point Nonlinear Equations Subdivided Manifold variable dimension algorithm

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A better triangulation for Wright's 2(n)-ray algorithm

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 1997年第2期212卷 417-429页

作者： Dang, CY Talman, D TILBURG UNIV DEPT ECONNL-5000 LE TILBURGNETHERLANDS

We give a triangulation for Wright's 2(n)-ray algorithm to compute solutions of nonlinear equations. According to measures of efficiency of triangulations, it is better than any other available triangulation for t... 详细信息

关键词： variable dimension algorithm Fixed-point algorithms N-cube Extra dimension algorithms nonlinear equations Wright stain N-cube triangulation

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A simplicial approach to the determination of an integer point of a simplex

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MATHEMATICS OF OPERATIONS RESEARCH 1998年第2期23卷 403-415页

作者： Dang, CY Van Maaren, H Aukland Univ Dept Engn Sci Auckland New Zealand Delft Univ Technol Fac Tech Math & Informat NL-2600 AA Delft Netherlands

We propose an algorithm to find an integer point of a simplex. The algorithm is based on an integer labeling rule and a triangulation of the space. Starting atom integer point, the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension. Within a finite number of iterations, the algorithm either yields an integer point of the simplex or proves that no such point exists.

关键词： integer point simplex integer programming integer labeling triangulation variable dimension algorithm

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THE OCTAHEDRAL algorithm, A NEW SIMPLICIAL FIXED-POINT algorithm

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MATHEMATICAL PROGRAMMING 1981年第1期21卷 47-69页

作者： WRIGHT, AH Department of Mathematics Western Michigan University 49008 Kalamazoo Michigan USA

A new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented. It uses the restrart technique of Merrill to improve the accuracy of the solution. The algorithm is shown to converge quadratically under certain conditions. The algorithm should be efficient and relatively easy to implement.

关键词： Fixed Point algorithm Computation of Fixed Points Nonlinear Equations variable dimension algorithm

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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 variable dimension algorithm until an approximate solution is found on K x variable dimension algorithm. 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 variable dimension algorithm, the variable dimension algorithm is again used until a new approximate solution is found on K x variable dimension algorithm. 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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AN EFFICIENT IMPLEMENTATION OF MERRILL'S METHOD FOR SPARSE OR PARTIALLY SEPARABLE SYSTEMS OF NONLINEAR EQUATIONS

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SIAM JOURNAL ON OPTIMIZATION 1991年第2期1卷 206-221页

作者： Bing, Yang Lin, Gao Harbin Shipbuilding Engn Inst Dept Engn Management Harbin 150001 Peoples R China Harbin Shipbuilding Engn Inst Dept Math & Mech Harbin 150001 Peoples R China

When Merrill's method without an extra dimension is applied to solving sparse or partially separable systems of nonlinear equations, the computational efficiency can be further improved, i.e., a larger piece of linearity can be traversed in one step by using a suitable linear system. One of the linear systems is updated and the corresponding technique is shown to update information about the linear system and the large piece in the implementation of the method. Some numerical results of the method support the claim that it is efficient. Also, some mistakes from a previous paper, in which the main technique exploiting sparsity was proposed, are corrected.

关键词： fixed point algorithm systems of nonlinear equations variable dimension algorithm sparsity partial separability

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On the existence and approximation of zeroes

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Mathematical Programming 1984年第1期28卷 1-24页

作者： van der Laan, Gerard Department of Actuarial Sciences and Econometrics Free University Amsterdam De Boelelaan 1081 Netherlands

In this paper we consider the problem of finding zeroes of a continuous function f from a convex, compact subset U of ℝn to ℝn. In the first part of the paper it is proved that f has a computable zero if f:Cn→ℝn sati... 详细信息

In this paper we consider the problem of finding zeroes of a continuous function f from a convex, compact subset U of ℝⁿ to ℝⁿ. In the first part of the paper it is proved that f has a computable zero if f:Cⁿ→ℝⁿ satisfies the nonparallel condition for any two antipodal points on bd Cⁿ, i.e. if for any x∈bd Cⁿ, f(x)≠αf(-x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bd Cⁿ to the face -τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes of f. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extending f:U→ℝⁿ to a function g from a cube containing U to ℝⁿ, the procedure can be applied to any continuous function f without having any information about the global and local Brouwer degrees a priori. © 1984 The Mathematical Programming Society Inc.

关键词： Complementary 1-Simplex Simplical Approximation variable dimension algorithm

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A New Arbitrary Starting Simplicial algorithm for Computing an Integer Point of an n-dimensional Simplex

A New Arbitrary Starting Simplicial Algorithm for Computing ...

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第六届运筹学及其应用国际研讨会

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

Determining whether there is an integer point in an n-dimensional simplex is an NPcomplete problem. In this paper, a new arbitrary starting variable dimension algorithm is developed for computing an integer point of an n-dimensional simplex. The algorithm is derived from an introduction of an integer labeling rule and an application of a triangulation of the space and is composed of two phases, one of which forms a variable dimension algorithm and the other a fulldimension pivoting procedure. Starting at an arbitrary integer point, the algorithm interchanges from one phase to the other if necessary and follows a finite simplicial path that either leads to an integer point of the simplex or proves that no such points exist. An advantage of the algorithm is that all the existing superior triangulations can be its underlying triangulations without any modification.

关键词： integer point simplex Diophantine equation nonnegative integer solution unimodular transformation integer programming integer labeling triangulation variable dimension algorithm pivoting procedure

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A new algorithm to solve calculus of variations problems using Wolfe's convergence theory, part II: Implementation

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Optimization 1996年第3期37卷 267-277页

作者： Zhou, W. Chalabi, Z.S. Process Systems Engineering Silsoe Research Institute Silsoe Bedford MK45 4HS Wrest Park United Kingdom

In a companion paper (A new algorithm to solve calculus of variations problems using Wolfe's convergence theory, Part I: Theory and algorithm), a new algorithm with the properties of guaranteed finite termination and global convergence was proposed to solve calculus of variations problems. A detailed implementation of the algorithm is presented here. The implementation proposes a variable dimension method and incorporates a stable procedure to solve for the descent search direction. Numerical examples are given to illustrate the algorithm.

关键词： Calculus of variations Descent algorithm Finite termination Global convergence Stability variable dimension algorithm

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