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  • 1
    Electronic Resource
    Electronic Resource
    An improvement of fixed point algorithms by using a good triangulation (1980)
    Springer
    Mathematical programming 18 (1980), S. 274-285 
    Details
    ISSN: 1436-4646
    Keywords: Triangulation ; Average Directional Density ; Fixed Point ; Equilateral Triangles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider measures for triangulations ofR n. A new measure is introduced based on the ratio of the length of the sides and the content of the subsimplices of the triangulation. In a subclass of triangulations, which is appropriate for computing fixed points using simplicial subdivisions, the optimal one according to this measure is calculated and some of its properties are given. It is proved that for the average directional density this triangulation is optimal (within the subclass) asn goes to infinity. Furthermore, we compare the measures of the optimal triangulation with those of other triangulations. We also propose a new triangulation of the affine hull of the unit simplex. Finally, we report some computational experience that confirms the theoretical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01588323
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A class of simplicial restart fixed point algorithms without an extra dimension (1981)
    Springer
    Mathematical programming 20 (1981), S. 33-48 
    Details
    ISSN: 1436-4646
    Keywords: Triangulation ; Fixed Point ; Labelling ; Approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices. Ideas of that paper are used to construct a class of triangulations ofR n. More precisely, for somek, 1 ≤k ≤ n, and positive integersm 1 ⋯ , mk with sumn, a triangulation ofR n is obtained by triangulating the cells which are formed by taking the product of given triangulations ofR mj, j = 1, ⋯ ,k. The triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inR n, being the starting point of the algorithm. Fork = n we obtain theK′ triangulation originally due to Todd. Each element of the class can be used to find a simplex which approximates a fixed point of a mapping onR n by generating a unique path of adjacent simplices of variable dimension starting with the pointv. We also give convergence conditions. It is indicated how in casek = n a connected set of fixed points can be generated. Moreover, we give some computational experience.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589331
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A restart algorithm for computing fixed points without an extra dimension (1979)
    Springer
    Mathematical programming 17 (1979), S. 74-84 
    Details
    ISSN: 1436-4646
    Keywords: Fixed Points ; Triangulation ; Grid Size ; Labelling Rule
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm to compute a fixed point of an upper semicontinuous point to set mapping using a simplicial subdivision is introduced. The new element of the algorithm is that for a given grid it does not start with a subsimplex but with one (arbitrary) point only; the algorithm will terminate always with a subsimplex. This subsimplex yields an approximation of a fixed point and provides the starting point for a finer grid. Some numerical results suggest that this algorithm converges more rapidly than the known algorithms. Moreover, it is very simple to implement the algorithm on the computer.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01588226
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  • 4
    Electronic Resource
    Electronic Resource
    A new subdivision for computing fixed points with a homotopy algorithm (1980)
    Springer
    Mathematical programming 19 (1980), S. 78-91 
    Details
    ISSN: 1436-4646
    Keywords: Triangulation ; Homotopy Function ; Fixed Point ; Grid Refinement
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper a triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inR n . This triangulation allows for factors of incrementation of more than two. The factor may be of any size and even different at each level. Also the starting point on a new level may be any gridpoint of the last found completely labelled subsimplex on the last level. So, the decision which new factor of incrementation and which starting point is used, can be made on the ground of previous approximations. Doing so, the convergence rate can be accelerated without using restart methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581629
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Boekbesprekingen - Reviews (1987)
    Springer
    De economist 135 (1987), S. 237-269 
    Details
    ISSN: 1572-9982
    Source: Springer Online Journal Archives 1860-2000
    Topics: Economics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01718265
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  • 6
    Electronic Resource
    Electronic Resource
    Complementary pivoting on a pseudomanifold structure with applications in the decision sciences (1988)
    Springer
    Acta applicandae mathematicae 11 (1988), S. 99-101 
    Details
    ISSN: 1572-9036
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00047116
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  • 7
    Electronic Resource
    Electronic Resource
    A new variable dimension simplicial algorithm for computing economic equilibria onS n ×R + m1 (1995)
    Springer
    Journal of optimization theory and applications 87 (1995), S. 679-701 
    Details
    ISSN: 1573-2878
    Keywords: Economic equilibria ; stationary point problems ; simplicial algorithms ; simplicial subdivisions ; adjustment processes ; piecewise linear approximations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new variable-dimension simplicial algorithm is developed to compute economic equilibria on the Cartesian product of then-dimensional unit price simplexS n and them-dimensional production activity spaceR + m . The algorithm differs from other algorithms in the number of directions in which the algorithm may leave the starting point. More precisely, the algorithm has 2 n+m+1 –2 rays to leave the starting point, whereas the other algorithms have at most 2 m (n+1) rays. The path of points generated by the algorithm can be interpreted as a globally and universally convergent price and production adjustment process. The process as well as the convergence condition are much more natural and economically meaningful than the adjustment processes obtained by other simplicial algorithms. Furthermore, we apply the algorithm to economies with linear production, economies with constant returns to scale, and economies with increasing returns to scale.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192139
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  • 8
    Electronic Resource
    Electronic Resource
    A procedure for finding Nash equilibria in bi-matrix games (1991)
    Springer
    Mathematical methods of operations research 35 (1991), S. 27-43 
    Details
    ISSN: 1432-5217
    Keywords: bi-matrix game ; Nash equilibrium ; complementarity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung In diesem Beitrag entwickeln wir ein neues Verfahren zur Berechnung eines Nash-Gleichgewichtspunktes für Zweimatrizen-Spiele. Das Standardverfahren zur Lösung dieser Spiele ist der Lemke-Howson Algorithmus. Dieses Pivotverfahren löst ein lineares Komplementaritätsproblem. Da dieses Verfahren nur in einer beschränkten Anzahl Punkte beginnen kann, können nicht alle Gleichgewichtspunkte berechnet werden. Unser Verfahren löst ein stationäres Punktproblem definiert auf die Strategiemenge und darf in einem beliebigen Punkt angefangen werden. Der Weg der Punkte des Algorithmus hat eine einladende spieltheoretische Interpretation. Wenn im Startpunkt alle Strategien mit positiver Wahrscheinlichkeit gewählt werden, dann ist das gefundene Nash-Gleichgewicht perfekt. Auch dürfen alle Gleichgewichtspunkte berechnet werden durch mehrere Startpunkte zu wählen.
    Notes: Abstract In this paper we consider the computation of Nash equilibria for noncooperative bi-matrix games. The standard method for finding a Nash equilibrium in such a game is the Lemke-Howson method. That method operates by solving a related linear complementarity problem (LCP). However, the method may fail to reach certain equilibria because it can only start from a limited number of strategy vectors. The method we propose here finds an equilibrium by solving a related stationary point problem (SPP). Contrary to the Lemke-Howson method it can start from almost any strategy vector. Besides, the path of vectors along which the equilibrium is reached has an appealing game-theoretic interpretation. An important feature of the algorithm is that it finds a perfect equilibrium when at the start all actions are played with positive probability. Furthermore, we can in principle find all Nash equilibria by repeated application of the algorithm starting from different strategy vectors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01415958
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  • 9
    Electronic Resource
    Electronic Resource
    Intersection Theorems with a Continuum of Intersection Points (1998)
    Springer
    Journal of optimization theory and applications 96 (1998), S. 311-335 
    Details
    ISSN: 1573-2878
    Keywords: Intersection theorems ; zero-point problems ; economic equilibrium ; connectedness ; closed coverings ; balancedness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In all existing intersection theorems, conditions are given under which a certain subset of a collection of sets has a nonempty intersection. In this paper, conditions are formulated under which the intersection is a continuum of points satisfying some interesting topological properties. In this sense, the intersection theorems considered in this paper belong to a new class. The intersection theorems are formulated on the unit cube and it is shown that both the vector of zeroes and the vector of ones lie in the same component of the intersection. An interesting application concerns the model of an economy with price rigidities. Using the intersection theorems of this paper, it is easily shown that there exists a continuum of zero points in such a model. The intersection theorems treated give a generalization of the well-known lemmas of Knaster, Kuratowski, and Mazurkiewicz (Ref. 1), Scarf (Ref. 2), Shapley (Ref. 3), and Ichiishi (Ref. 4). Moreover, the results can be used to sharpen the usual formulation of the Scarf lemma on the cube.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022618130893
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Simplicial zero-point algorithms: A unifying description (1986)
    Springer
    Journal of optimization theory and applications 50 (1986), S. 165-182 
    Details
    ISSN: 1573-2878
    Keywords: Zero points ; stationary points ; simplicial algorithms ; limiting paths ; Merrill's condition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00938483
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    Paper (German National Licenses)
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