Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods
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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.
- E. Allgower and K. Georg, “Simplicial and continuation methods for approximating fixed points and solutions to systems of equations”,SIAM Review 22 (1980) 28–85.
- R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”,Linear Algebra and Its Applications 1 (1968) 103–125.
- G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).
- B.C. Eaves, “A short course in solving equations with PL homotopies”,SIAM-AMS Proceedings 9 (1976) 73–143.
- 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.
- B.C. Eaves and H. Scarf, “The solution of systems of piecewise linear equations”,Mathematics of Operations Research 1 (1976) 1–27.
- R. Freund, “Variable-dimension complexes with applications”, Ph.D. Dissertation, Stanford University (Stanford, CA, June 1980).
- L. van der Heyden, “Restricted primitive sets in a regularly distributed list of vectors and simplicial subdivisions with arbitrary refinement factors”, Discussion Paper Series No. 79D, John Fitzgerald Kennedy School of Government, Harvard University (Cambridge, MA, January 1980).
- M. Kojima, “Studies on piecewise-linear approximation of piecewise-C 1 mappings in fixed points and complementarity theory”,Mathematics of Operations Research 3 (1978) 17–36.
- M. Kojima, “A note on ‘a new algorithm for computing fixed points’ by van der Laan and Talman”, in: W. Forster, ed.,Numerical solution on highly nonlinear problems (North-Holland, New York, 1980) pp. 37–42.
- M. Kojima, “An introduction to variable dimension algorithms for solving systems of equations”, in: E.L. Allgower, K. Glashoff and H.-O. Peitgen, eds.,Numerical solution of nonlinear equations (Springer, Berlin, 1981) pp. 199–237.
- G. van der Laan, “Simplicial fixed point algorithms”, Ph.D. Dissertation, Free University (Amsterdam, 1980).
- G. van der Laan and A.J.J. Talman, “On the computation of fixed points in the product space of the unit simplices and an application to non cooperativen-person games”, Free University (Amsterdam, October 1978).
- G. van der Laan and A.J.J. Talman, “A class of simplicial subdivisions for restart fixed point algorithms without an extra dimension”,Mathematical Programming 20 (1981) 33–48.
- 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, New York, 1980) pp. 3–36.
- G. van der Laan and A.J.J. Talman, “A restart algorithm for computing fixed points without 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-Verlag, Berlin, 1979) pp. 247–256.
- G. van der Laan and A.J.J. Talman, “A new subdivision for computing fixed points with a homotopy algorithm”, Free University (Amsterdam, February 1979).
- C.E. Lemke and J.T. Howson Jr., “Equilibrium points of bimatrix games”,SIAM Journal on Applied Mathematics 12 (1964) 413–423.
- O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain upper semi-continuous point to set mappings”, Ph.D. Dissertation, University of Michigan (Ann Arbor, MI, 1972).
- S. Mizuno, “A simplicial algorithm for finding all solutions to polynomial systems of equations”, Master Thesis, Department of System Sciences, Tokyo Institute of Technology (Tokyo, February 1981).
- P.M. Reiser, “A modified integer labelling for complementarity algorithms”,Mathematics of Operations Research 6 (1981) 129–139.
- C.P. Rourke and B.J. Sanderson,Introduction to piecewise-linear topology (Springer-Verlag, New York, 1972).
- R. Saigal, “Fixed point computing methods”, in: A.G. Holzman, ed.,Operations research support methodology (Marcel Dekker, New York, 1979) pp. 545–566.
- H. Scarf, “The approximation of fixed points of a continuous mapping”,SIAM Journal on Applied Mathematics 15 (1967) 1328–1343.
- S. Shamir, “A homotopy fixed point algorithm with an arbitrary integer refinement factor”, Department of Engineering-Economic Systems, Stanford University (Stanford, CA, May 1979).
- A.J.J. Talman, “Variable dimension fixed point algorithms and triangulations”, Ph.D. Dissertation, Free University (Amsterdam, 1980).
- M.J. Todd,The computation of fixed points and applications (Springer-Verlag, New York, 1976).
- M.J. Todd, “Exploiting structure in fixed-point computation”,Mathematical Programming 18 (1980) 233–247.
- M.J. Todd, “Fixed-point algorithm that allow restarting without an extra dimension”, Technical Report No. 379, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, September 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, New York, 1980) pp. 43–69.
- M.J. Todd, “Traversing large large pieces of linearity in algorithms that solve equation by following piecewise-linear paths”,Mathematics of Operations Research 5 (1980) 242–257.
- M.J. Todd and A.H. Wright, “A variable-dimension simplicial algorithm for antipodal fixedpoint theorems”,Numerical Functional Analysis and Optimization 2 (1980) 155–186.
- A.H. Wright, “The octahedral algorithm, a new simplicial fixed point algorithm”,Mathematical Programming 21 (1981) 47–69.
- Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods
Volume 24, Issue 1 , pp 177-215
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