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  • 1
    Electronic Resource
    Electronic Resource
    A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes (1993)
    Springer
    Mathematical programming 61 (1993), S. 233-249 
    Details
    ISSN: 1436-4646
    Keywords: Minimum norm point ; minimum distance pair of points ; recursive algorithm ; convex quadratic program
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract For a given pair of finite point setsP andQ in some Euclidean space we consider two problems: Problem 1 of finding the minimum Euclidean norm point in the convex hull ofP and Problem 2 of finding a minimum Euclidean distance pair of points in the convex hulls ofP andQ. We propose a finite recursive algorithm for these problems. The algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582149
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A continuous deformation algorithm for variational inequality problems on polytopes (1994)
    Springer
    Mathematical programming 64 (1994), S. 103-122 
    Details
    ISSN: 1436-4646
    Keywords: Variational inequality ; Stationary point ; Simplicial algorithm ; Variable dimension algorithm ; Continuous deformation ; Triangulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A continuous deformation algorithm is proposed for solving a variational inequality problem on a polytopeK. The algorithm embeds the polytopeK intoK× [0, ∞) and starts by applying a variable dimension algorithm onK× {0} until an approximate solution is found onK× {0}. Then by tracing the path of solutions of a system of equations the algorithm virtually follows a path of approximate solution inK× [0, ∞). When the path inK× [0, ∞) returns to level 0, i.e.,K× {0}, the variable dimension algorithm is again used until a new approximate solution is found onK× {0}. The setK× [0, ∞) 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582567
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods (1982)
    Springer
    Mathematical programming 24 (1982), S. 177-215 
    Details
    ISSN: 1436-4646
    Keywords: Variable Dimension Algorithm ; Fixed Point ; Subdivided Manifold ; Nonlinear Equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585103
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A path-following procedure to find a proper equilibrium of finite games (1993)
    Springer
    International journal of game theory 22 (1993), S. 249-259 
    Details
    ISSN: 1432-1270
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We propose a procedure to find a proper equilibrium of finiten-person games, which was introduced by Myerson as a refinement of perfect equilibrium. The procedure is a new variable dimension fixed point algorithm having Π i =1/n m i ! directions in which it may leave the starting point, wherem i is the number of thei-th player's pure strategies.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01240056
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    A Finite Algorithm for Globally Optimizing a Class of Rank-Two Reverse Convex Programs (1998)
    Springer
    Journal of global optimization 12 (1998), S. 247-265 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; Reverse convex program ; Rank-two quasiconcave function ; Parametric simplex algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we propose an algorithm for solving a linear program with an additional rank-two reverse convex constraint. Unlike the existing methods which generate approximately optimal solutions, the algorithm provides a rigorous optimal solution to this nonconvex problem by a finite number of dual pivot operations. Computational results indicate that the algorithm is practical and can solve fairly large scale problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008216024699
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    The (n + 1)2 m -ray algorithm: a new simplicial algorithm for the variational inequality problem on ℝ + m ×S n (1993)
    Springer
    Annals of operations research 44 (1993), S. 93-113 
    Details
    ISSN: 1572-9338
    Keywords: Variational inequality problem ; nonlinear complementarity problem ; simplicial algorithm ; triangulation ; piecewise linear approximation ; convergence condition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper we propose a variable dimension simplicial algorithm for solving the variational inequality problem on the cross product of the nonnegative orthant ℝ + m of them-dimensional Euclidean space ℝ m and then-dimensional unit simplexS n of ℝ n+1. Starting from an arbitrary point (u, v) єℝ + m ×S n, the algorithm generates a piecewise linear path in ℝ + m ×S n. The path is traced by making alternately linear programming pivot operations and replacement steps in an appropriate simplicial subdivision of ℝ + m ×S n. The algorithm differs from the thus far known algorithm in the number of directions in which it may leave the starting point. More precisely, the algorithm has (n+1)2 m rays to leave the starting point whereas the existing algorithm hasn+m+1 rays. A convergence condition is presented and the accuracy estimation of an approximate solution generated is also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02073592
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    Paper (German National Licenses)
    Fulltext
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