Abstract
We present an efficient simplicial algorithm for computing a zero of a point-to-set mapping that is formed by piecing together smooth functions. Such mappings arise in nonlinear programming and economic equilibrium problems. Our algorithm, under suitable regularity conditions on the problem, generates a sequence converging at least Q-superlinearly to a zero of the mapping. Asymptotically, it operates in a space of reduced dimension, analogous to an active set strategy in the optimization setting, but it switches active sets automatically. Results of computational experiments are given.
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References
E. Allgower and K. Georg, “Simplicial and continuation methods for approximating fixed points”,SIAM Review, 22 (1980) 28–85.
M. Avriel,Nonlinear programming: Analysis and methods (Prentice-Hall, Englewood Cliffs, NJ, 1976).
S.A. Awoniyi, “A piecewise-linear homotopy algorithm for computing zeros of certain point-to-set mappings”, Ph.D. Thesis, Cornell University, Ithaca, NY (1980).
L.J. Billera, “Some theorems on the core of ann-person game without side payments”,SIAM Journal on Applied Mathematics 18 (1970) 567–579.
C.G. Broyden, “A class of methods for solving nonlinear simultaneous equations”,Mathematics of Computation 19 (1965) 577–593.
A.R. Colville, “A comparative study of nonlinear programming codes”, Tech. Rept. No. 320-2949, IBM New York Scientific Center, New York (1968).
B.C. Eaves, “A short course in solving equations with PL homotopies”, in: R.W. Cottle and C.E. Lemke, eds.,Nonlinear Programming, Proceedings of the Ninth SIAM-AMS Symposium in Applied Mathematics (SIAM, Philadelphia, PA, 1976) pp. 73–143.
M.L. Fisher and F.J. Gould, “A simplicial algorithm for the nonlinear complementarity problem”,Mathematical Programming, 6 (1974) 281–300.
T. Hansen and T.C. Koopmans, “Definition and computation of a capital stock invariant under optimization”, in: R.H. Day and S.M. Robinson, eds.,Mathematical topics in economic theory and computation (SIAM, Philadelphia, PA, 1972) pp. 113–121.
T.J. Kehoe, “An index theorem for general equilibrium models with production”,Econometrica 45 (1980) 1211–1232.
M. Kojima, “Computational methods for solving the nonlinear complementarity problem”,Keio Engineering Reports 27 (1974) 1–41.
M. Kojima, “On the homotopic approach to systems of equations with separable mappings”,Mathematical Programming Study 7 (1978) 170–184.
G. van der Laan and A.J.J. Talman, “A class of simplicial restart fixed point algorithms without an extra dimension”,Mathematical Programming 20 (1981) 33–48.
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, Department of Industrial Engineering, University of Michigan, Ann Arbor, MI (1972).
M.J.D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations”, in: G.A. Watson, ed.,Proceedings Dundee Conference on Numerical Analysis. Lecture Notes in Mathematics 630 (Springer, Berlin, 1977) pp. 144–157.
R. Saigal, “On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting”,Mathematics of Operations Research 2 (1977) 108–124.
H. Scarf, “The approximation of fixed points of a continuous mapping”,SIAM Journal on Applied Mathematics 15 (1967), 1328–1343.
H.E. Scarf and T. Hansen,Computation of economic equilibria (Yale University Press, New Haven, CT, 1973).
L.S. Shapley, “On balanced games without side payments”, in: T.C. Hu and S.M. Robinson, eds.,Mathematical Programming (Academic Press, New York, 1973) pp. 261–290.
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, “A note on computing equilibria in economies with activity analysis models of production”,Journal of Mathematical Economics 6 (1979) 135–144.
M.J. Todd, “A quadratically-convergent fixed-point algorithm for economic equilibria and linearly constrained optimization”,Mathematical Programming 18 (1980) 111–126.
M.J. Todd, “Traversing large pieces of linearity in algorithms that solve equations by following piecewise-linear paths”,Mathematics of Opertions Research 5 (1980) 242–257.
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. 43–69.
M.J. Todd, “PLALGO: a Fortran implementation of a piecewise-linear homotopy algorithm for solving systems of nonlinear equations”, Tech. Rep. No. 454, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY (1980).
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Research of this author was supported by a Fellowship from the Rockeffeller Foundation.
Research of this author was partially supported by a fellowhip from the John Simon Guggenheim Memorial Foundation and by National Science Foundation Grant ECS-7921279.
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Awoniyi, S.A., Todd, M.J. An efficient simplicial algorithm for computing a zero of a convex union of smooth functions. Mathematical Programming 25, 83–108 (1983). https://doi.org/10.1007/BF02591720
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DOI: https://doi.org/10.1007/BF02591720