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.
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References
B.C. Eaves, “Homotopies for computation of fixed points”,Mathematical Programming 3 (1972) 1–22.
B.C. Eaves and R. Saigal, “Homotopies for computation of fixed points on unbounded regions”,Mathematical Programming 3 (1972) 225–237.
W. Forster, “Fixed point algorithms: background and estimates for implementation on array processors”, Faculty of Mathematical Studies, Preprint Series No. 9, University of Southampton, England (1978).
R. Kellogg, T.-Y. Li and J. Yorke, “A constructive proof of Brouwer fixed point theorem and computational results”,SIAM Journal on Numerical Mathematics 13 (1976) 473–483.
H.W. Kuhn and J.G. MacKinnon, “Sandwich method for finding fixed points”,Journal of Optimization Theory and Applications 17 (1975) 189–204.
G. van der Laan, “Simplicial fixed point algorithms”, Dissertation, Vrije Universiteit, Amsterdam, The Netherlands (1980).
G. van der Laan and A.J.J. Talman, “A restart algorithm for computing fixed points without an extra dimension”,Mathematical Programming 17 (1979) 74–84.
G. van der Laan and A.J.J. Talman, “A restart algorithm without an artificial level for computing fixed points on unbounded regions”, in: H.-O. Peitgen and H.-O. Walther, eds.,Functional differential equations and approximation of fixed points, Lecture Notes in Mathematics 730 (Springer, Berlin, 1979) pp. 247–256.
G. van der Laan and A.J.J. Talman, “An improvement of fixed point algorithms by using a good triangulation”,Mathematical Programming 18 (1980) 274–285.
G. van der Laan and A.J.J. Talman, “On the computation of fixed points in the product space of unit simplices and an application to noncooperativeN person games”, Interfaculteit der Actuariële Wetenschappen en Econometrie, Onderzoekverslag 35, Vrije Universiteit, Amsterdam, The Netherlands (1978).
G. van der Laan and A.J.J. Talman, “A new subdivision for computing fixed points with a homotopy algorithm”,Mathematical Programming 19 (1980) 78–91.
G. van der Laan and A.J.J. Talman, “Interpretation of the variable dimension fixed point algorithm with an artificial level”, Interfaculteit der Actuariële Wetenschappen en Econometrie, Onderzoekverslag 47, Vrije Universiteit, Amsterdam, The Netherlands (1979).
G. van der Laan and A.J.J. Talman, “Convergence and properties of recent variable dimension algorithms”, in: W. Forster, ed.,Numerical solution of highly nonlinear problems (North-Holland, Amsterdam, 1980) pp. 3–36.
O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain upper semi-continuous point to set mappings”, Ph.D. Thesis, University of Michigan, Ann Arbor, MI (1972).
P.M. Reiser, “A modified integer labelling for complementarity algorithms”, Institut für Operations Research, Universität Zürich, Zürich, Switzerland (1978).
P.M. Reiser, “Ein hybrides Verfahren zur Lösung von nichtlinearen Komplementaritätsproblemen und seine Konvergenzeigenschaften”, Dissertation, Eidgenössischen Technischen Hochschule, Zürich, Switzerland (1978).
R. Saigal, “On the convergence of algorithms for solving equations that are based on methods of complementary pivoting”,Mathematics of Operations Research 2 (1977) 108–124.
H.E. Scarf, “The approximation of fixed points of a continuous mapping”,SIAM Journal on Applied Mathematics 15 (1967) 1328–1343.
H.E. Scarf,The computation of economic equilibria (Yale University Press, New Haven, CT, 1973).
A.J.J. Talman, “Variable dimension fixed point algorithms and triangulations”, Dissertation, Vrije Universiteit, Amsterdam, The Netherlands (1980).
M.J. Todd,The computation of fixed points and applications (Springer, Berlin, 1976).
M.J. Todd, “Improving the convergence of fixed point algorithms”,Mathematical Programming Study 7 (1978) 151–169.
M.J. Todd, “Fixed-point algorithms that allow restarting without an extra dimension”, School of Operations Research and Industrial Engineering, Tech. Rept. No. 379, Cornell University Ithaca, NY (1978).
M.J. Todd, “Global and local convergence and monotonicity results for a recent variabledimension simplicial algorithm”, in: W. Forster, ed.,Numerical solution of highly nonlinear problems (North-Holland, Amsterdam, 1980) pp. 43–70.
M.J. Todd and A.H. Wright, “A variable-dimension simplicial algorithm for antipodal fixed point theorems”, School of Operations Research and Industrial Engineering, Tech. Rept. No. 417, Cornell University, Ithaca, NY (1979).
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van der Laan, G., Talman, A.J.J. A class of simplicial restart fixed point algorithms without an extra dimension. Mathematical Programming 20, 33–48 (1981). https://doi.org/10.1007/BF01589331
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DOI: https://doi.org/10.1007/BF01589331