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Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimizations

  • Chuangyin Dang
Reference work entry

Abstract

Simplicial methods were originated by Scarf for approximating fixed points of continuous mappings. They have many applications in economics and science. With regard to its geometric structure, a simplicial method can be classified as either a variable dimension simplicial method or a homotopy simplicial method. A variable dimension simplicial method works directly on the interested space, whereas a simplicial homotopy method needs to introduce an extra dimension. It is well known that integer programming is equivalent to determining whether there is an integer point in a polytope. Simplicial methods were extended to computing an integer point in a polytope. There is a significant difference between simplicial methods for approximating fixed points and simplicial methods for integer programming, though they both have the same foundation. Two most important components of simplicial methods are labeling rules and triangulations. Efficiency of simplicial methods depends critically on the underlying triangulations. Three simplest triangulations of R n are the K 1-triangulation, the J 1-triangulation, and the D 1-triangulation. This chapter presents a brief introduction to these developments of simplicial methods.

Keywords

Extra Dimension Integer Point Homotopy Method Dimensional Simplex Unit Simplex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was partially supported by GRF: CityU 112809 of the Government of Hong Kong SAR.

Recommended Reading

  1. 1.
    E.L. Allgower, K. Georg, Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. SIAM Rev. 22, 28–85 (1980)MathSciNetMATHGoogle Scholar
  2. 2.
    E.L. Allgower, K. Georg, Piecewise linear methods for nonlinear equations and optimization. J. Comput. Appl. Math. 124, 245–261 (2000)MathSciNetMATHGoogle Scholar
  3. 3.
    K.J. Arrow, F.H. Hahn, General Competitive Analysis (Holden-Day, San Francisco 1971)MATHGoogle Scholar
  4. 4.
    I. Barang, Borsuk’s theorem through complementary pivoting. Math. Program. 18, 84–88 (1980)Google Scholar
  5. 5.
    M.N. Broadie, B.C. Eaves, A variable rate refining triangulation. Math. Program. 38, 161–202 (1987)MathSciNetMATHGoogle Scholar
  6. 6.
    P.S. Brooks, Infinite regression in the Eaves-Saigal algorithm. Math. Program. 19, 313–327 (1980)MathSciNetMATHGoogle Scholar
  7. 7.
    L.E. Brouwer, ÄUber Abbildung von Mannig-faltigkeiten, Mathematische Annalen 71, 97–115 (1912)MATHGoogle Scholar
  8. 8.
    A. Charnes, G.B. Carcia, C.E. Lemke, Constructive proofs of theorems relating to: F(x)=y. with applications. Math. Program. 12, 328–343 (1977)MATHGoogle Scholar
  9. 9.
    R.W. Cottle, Minimal triangulation of the 4-cube. Discret. Math. 40, 25–29 (1982)MathSciNetMATHGoogle Scholar
  10. 10.
    R.W. Cottle, G.B. Dantzig, Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)MathSciNetMATHGoogle Scholar
  11. 11.
    R.W. Cottle, C.E. Lemke, Nonlinear programming, SIAM-AMS Proceedings, vol. 9 (AMS, Providence, 1976)Google Scholar
  12. 12.
    Y. Dai, G. van der Laan, A.J.J. Talman, Y. Yamamoto, A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron. SIAM J. Optim. 1, 151–165 (1991)MathSciNetMATHGoogle Scholar
  13. 13.
    C. Dang, The D 1-triangulation of R n for simplicial algorithms for computing solutions of nonlinear equations. Math. Oper. Res. 16, 148–161 (1991)MathSciNetMATHGoogle Scholar
  14. 14.
    C. Dang, The \(D_{2}^{{\ast}}\)-triangulation for continuous deformation algorithms to compute solutions of nonlinear equations. SIAM J. Optim. 3, 784–799 (1993a)MathSciNetMATHGoogle Scholar
  15. 15.
    C. Dang, The D 2-triangulation for simplicial homotopy algorithms for computing solutions of nonlinear equations. Math. Program. 59, 307–324 (1993b)MathSciNetMATHGoogle Scholar
  16. 16.
    C. Dang, Triangulations and Simplicial Methods. Lecture Notes in Economics and Mathematical Systems, vol. 421 (Springer, Berlin/New York, 1995), 196pMATHGoogle Scholar
  17. 17.
    C. Dang, An arbitrary starting homotopy-like simplicial algorithm for computing an integer point in a class of polytopes, SIAM J. Discret. Math. 23, 609–633 (2009)MathSciNetMATHGoogle Scholar
  18. 18.
    C. Dang, H. van Maaren, A simplicial approach to the determination of an integral point of a simplex. Math. Oper. Res. 23, 403–415 (1998)MathSciNetMATHGoogle Scholar
  19. 19.
    C. Dang, H. van Maaren, An arbitrary starting variable dimension algorithm for computing an integer point of a simplex. Comput. Optim. Appl. 14, 133–155 (1999)MathSciNetMATHGoogle Scholar
  20. 20.
    C. Dang, H. van Maaren, Computing an integer point of a simplex with an arbitrary starting homotopy-like simplicial algorithm, J. Comput. Appl. Math. 129, 151–170 (2001)MathSciNetMATHGoogle Scholar
  21. 21.
    C. Dang, Y. Ye, Computing an integer point in a class of polytopes. Lect. Notes Oper. Res. 14, 258–263 (2011)Google Scholar
  22. 22.
    G.B. Dantzig, B.C. Eaves, Studies in Optimization 10 (Washington, DC, American Mathematical Society, 1974)Google Scholar
  23. 23.
    G.B. Dantzig, B.C. Eaves, D. Gale, An algorithm for a piecewise linear model of trade and production with negative prices and bankruptcy. Math. Program. 16, 150–169 (1979)MathSciNetGoogle Scholar
  24. 24.
    R.H. Day, S.M. Robinson, Mathematical Topics in Economic Theory and Computation (Academic, New York, 1972)MATHGoogle Scholar
  25. 25.
    G. Debueu, Theory of Value (Wiley, New York, 1959)Google Scholar
  26. 26.
    T.M. Doup, Simplicial Algorithms on the Simplotope. Lecture Notes in Economics and Mathematical Systems, vol. 318 (Springer, Berlin, 1988)MATHGoogle Scholar
  27. 27.
    T.M. Doup, A.J.J. Talman, A new variable dimension algorithm fo find equilibria on the product space of unit simplices. Math. Program. 37, 319–355 (1987a)MathSciNetMATHGoogle Scholar
  28. 28.
    T.M. Doup, A.J.J. Talman, The 2-ray algorithm for solving equilibrium problems on the unit simplex. Methods Oper. Res. 57, 269–285 (1987b)MathSciNetMATHGoogle Scholar
  29. 29.
    T.M. Doup, A.J.J. Talman, A continuous deformation algorithm on the product space of unit simplices. Math. Oper. Res. 12, 485–521 (1987c)MathSciNetMATHGoogle Scholar
  30. 30.
    T.M. Doup, A.H. van den Elzen, A.J.J Talman, Simplicial algorithms for solving the nonlinear complementarity problem on the simplotope, in The Computation and Modelling of Economic Equilibria, ed. by A.J.J. Talman, G. van Laan. Conributions to Economic Analysis, vol. 167 (North-Holland, Amsterdam, 1987a), pp. 205–230, 125–154Google Scholar
  31. 31.
    T.M. Doup, G. van der Laan, A.J.J. Talman, The (2n + 1-2)-ray algorithm: a new simplicial algorithm to compute economic equilibria. Math. Program. 39, 241–252 (1987b)MATHGoogle Scholar
  32. 32.
    B.C. Eaves, An odd theorem. Proc. Am. Math. Soc. 26, 509–513 (1970)MathSciNetMATHGoogle Scholar
  33. 33.
    B.C. Eaves, On the basic theory of complementarity. Math. Program. 1, 68–75 (1971a)MathSciNetMATHGoogle Scholar
  34. 34.
    B.C. Eaves, Computing Kakutani fixed points. SIAM J. Appl. Math. 21, 236–244 (1971b)MathSciNetMATHGoogle Scholar
  35. 35.
    B.C. Eaves, The linear complementarity problem. Manage. Sci. 17, 612–634 (1971c)MathSciNetMATHGoogle Scholar
  36. 36.
    B.C. Eaves, Homotopies for the computation of fixed points. Math. Program. 3, 1–22 (1972)MathSciNetMATHGoogle Scholar
  37. 37.
    B.C. Eaves, Solving regular piecewise linear convex equations. Math. Program. Study 1, 96–119 (1974a)MathSciNetGoogle Scholar
  38. 38.
    B.C. Eaves, Properly labeled simplexes, in Studies in Optimization 10, ed. by G.B. Dantzig, B.C. Eaves (Washington, DC, American Mathematical Society, 1974b), pp. 71–93Google Scholar
  39. 39.
    B.C. Eaves, A short course in solving equations with PL homotopies, in Nonlinear Programming, ed. by R.W. Cottle, C.E. Lemke, SIAM-AMS Proceedings, vol. 9 (AMS, Providence, 1976), pp. 73–143Google Scholar
  40. 40.
    B.C. Eaves, Computing stationary points. Math. Program. Study 7, 1–14 (1978)MathSciNetMATHGoogle Scholar
  41. 41.
    B.C. Eaves, Permutation congruent transformations of the Freudenthal triangulation with minimum surface density. Math. Program. 29, 77–99 (1984a)MathSciNetMATHGoogle Scholar
  42. 42.
    B.C. Eaves, A Course in Triangulations for Solving Equations with Deformations. Lecture Notes in Economics and Mathematical Systems, vol. 234 (Springer, Berlin, 1984b)MATHGoogle Scholar
  43. 43.
    B.C. Eaves, R. Saigal, Homotopies for the computation of fixed points on unbounded regions. Math. Program. 3, 225–237 (1972)MathSciNetMATHGoogle Scholar
  44. 44.
    B.C. Eaves, H. Scarf, The solution of systems of piecewise linear equations. Math. Oper. Res. 1, 1–27 (1976)MathSciNetMATHGoogle Scholar
  45. 45.
    B.C. Eaves, J.A. Yorke, Equivalence of surface density and average directional density. Math. Oper. Res. 9, 363–375 (1984)MathSciNetMATHGoogle Scholar
  46. 46.
    B.C. Eaves, F.J. Gould, H.-O. Peitgen, M.J. Todd, Homotopy Methods and Global Convergence. NATO Conference Series, vol. 13 (Plenum, New York, 1983)MATHGoogle Scholar
  47. 47.
    C.R. Engles, Economics equilibrium under deformation of the economy, in Analysis and Computation of Fixed Points, ed. by S. Robinson (Academic, New York, 1980), pp. 213–410Google Scholar
  48. 48.
    M.L. Fisher, F.J. Gould, A simplicial algorithm for the nonlinear complementarity problem. Math. Program. 6, 281–300 (1974)MathSciNetMATHGoogle Scholar
  49. 49.
    M.L. Fisher, F.J. Gould, W.J. Tolle, A new simplicial approximation algorithm with restarts: relations between convergence and labelling, in Fixed Points: Algorithms and Applications, ed. by S. Karamardian (Academic, New York, 1977), pp. 41–58Google Scholar
  50. 50.
    W. Forster, Numerical Solution of Highly Nonlinear Problems (North-Holland, Amsterdam, 1980)MATHGoogle Scholar
  51. 51.
    W. Forster, Computing “all” solutions of systems of polynomial equations by simplicial fixed point algorithms, in [154], 39–58 (1987)Google Scholar
  52. 52.
    H. Freudenthal, Simplizialzerlegungen von Beschrankter Flachheit. Ann. Math. 43, 580–582 (1942)MathSciNetMATHGoogle Scholar
  53. 53.
    R.W. Freund, Variable dimension complexes part I: basic theory. Math. Oper. Res. 9, 479–497 (1984a)MathSciNetMATHGoogle Scholar
  54. 54.
    R.W. Freund, Variable dimension complexes part II: a unified approach to some combinatorial lemmas in topology. Math. Oper. Res. 9, 498–509 (1984b)MathSciNetMATHGoogle Scholar
  55. 55.
    R.W. Freund, Combinatorial theorems on the simplotope that generalize results on the simplex and cube. Math. Oper. Res. 11, 169–179 (1986)MathSciNetMATHGoogle Scholar
  56. 56.
    R.W. Freund, M.J. Todd, A constructive proof of Tucker’s combinatorial lemma. J. Comb. Theory A 30, 321–325 (1981)MathSciNetMATHGoogle Scholar
  57. 57.
    T. Fujisawa, E. Kuh, Piecewise linear theory of nonlinear networks. SIAM J. Appl. Math. 22, 307–328 (1972)MathSciNetMATHGoogle Scholar
  58. 58.
    C.B. Garcia, A fixed point theorem including the last theorem of Poincaré. Math. Program. 8, 227–239 (1975)Google Scholar
  59. 59.
    C.B. Garcia, A hybrid algorithm for the computation of fixed points. Manage. Sci. 22, 606–613 (1976)MATHGoogle Scholar
  60. 60.
    C.B. Garcia, Computation of solutions to nonlinear equations under homotopy invariance. Math. Oper. Res. 2, 25–29 (1977)MathSciNetMATHGoogle Scholar
  61. 61.
    C.B. Garcia, F.J. Gould, A theorem on homotopy paths. Math. Oper. Res. 3, 282–289 (1978)MathSciNetMATHGoogle Scholar
  62. 62.
    C.B. Garcia, F.J. Gould, Scalar labelings for homotopy paths. Math. Program. 17, 184–197 (1979)MathSciNetMATHGoogle Scholar
  63. 63.
    C.B. Garcia, F.J. Gould, Relations between several path following algorithms and local and global Newton methods. SIAM Rev. 22, 263–274 (1980)MathSciNetMATHGoogle Scholar
  64. 64.
    C.B. Garcia, T.Y. Li, On the number of solutions to polynomial systems of equations. SIAM Numer. Anal. 17, 540–546 (1980)MathSciNetMATHGoogle Scholar
  65. 65.
    C.B. Garcia, W.I. Zangwill, Determining all solutions to certain systems of nonlinear equations. Math. Oper. Res. 4, 1–14 (1979a)MathSciNetMATHGoogle Scholar
  66. 66.
    C.B. Garcia, W.I. Zangwill, Finding all solutions to polynomial systems and other systems of equations. Math. Program. 16, 159–176 (1979b)MathSciNetMATHGoogle Scholar
  67. 67.
    C.B. Garcia, W.I. Zangwill, An approach to homotopy and degree theory. Math. Oper. Res. 4, 390–405 (1979c)MathSciNetMATHGoogle Scholar
  68. 68.
    C.B. Garcia, W.I. Zangwill, A flex simplicial algorithm, in Numerical Solution of Highly Nonlinear Problems ed. by W. Forster (North-Holland, Amsterdam, 1980), pp. 71–92Google Scholar
  69. 69.
    C.B. Garcia, W.I. Zangwill, Pathways to Solutions, Fixed Points, and Equilibria. Series in Computational Mathematics (Prentice-Hall, Eanglewood Cliffs, 1981)MATHGoogle Scholar
  70. 70.
    K. Georg, An application of simplicial algorithms to variational inequalities, in Functional Differential Equations and Approximation of Fixed Points, ed. by H.O. Peitgen. Lecture Notes in Mathematics, vol. 730 (Springer, New York/Berlin, 1979), pp. 126–135Google Scholar
  71. 71.
    F.J. Gould, J.W. Tolle, A unified approach to complementarity in optimization. Discret. Math. 7, 225–271 (1974)MathSciNetMATHGoogle Scholar
  72. 72.
    F.J. Gould, J.W. Tolle, An existence theorem for solutions to f(x)=0. Math. Program. 11, 252–262 (1976)MathSciNetGoogle Scholar
  73. 73.
    F.J. Gould, J.W. Tolle, Complementary Pivoting on a Pseudomanifold Structure with Applications in the Decision Sciences. Sigma Series in Applied Mathematics, vol. 2 (Heldermann, Berlin, 1983)Google Scholar
  74. 74.
    T. Hansen, On the approximation of a competitive equilibrium. Ph.D. Thesis, Department of Economics, Yale University, New Haven (1968)Google Scholar
  75. 75.
    T. Hansen, On the approximation of Nash equilibrium points in an N-person noncooperative game. SIAM J. Appl. Math. 26, 622–637 (1974)MathSciNetMATHGoogle Scholar
  76. 76.
    M.W. Hirsch, A proof of the nonretractability of a cell onto its boundary. Proc. Am. Math. Soc. 14, 364–365 (1963)MATHGoogle Scholar
  77. 77.
    M.W. Hirsch, On algorithms for solving f(x)=0. Commun. Pure Appl. Math. 32, 281–312 (1979)MATHGoogle Scholar
  78. 78.
    M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, New York, 1984)Google Scholar
  79. 79.
    M. Hofkes, A simplicial algorithm to solve the nonlinear complementarity problem on S n ×\(R_{+}^{n}\). J. Optim. Theory Appl. 67, 551–565 (1990)MathSciNetMATHGoogle Scholar
  80. 80.
    T.C. Hu, S.M. Robinson, Mathematical Programming (Academic, New York, 1980)Google Scholar
  81. 81.
    M.M. Jeppson, A search for the fixed points of a continuous mapping, in Mathematical Topics in Economic Theory and Computation, ed. by R.H. Day, S.M. Robinson (Academic, New York, 1972), pp. 122–129Google Scholar
  82. 82.
    K. John, Parametric fixed point algorithms with applications to economic policy analysis. Comput. Oper. Res. 11, 157–178 (1984)MathSciNetMATHGoogle Scholar
  83. 83.
    S. Kakutani, A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)MathSciNetGoogle Scholar
  84. 84.
    K. Kamiya, A.J.J. Talman, Simplicial algorithm to find zero points of a function with special structure on a simplotope. Math. Oper. Res. 16, 609–626 (1991a)MathSciNetMATHGoogle Scholar
  85. 85.
    K. Kamiya, A.J.J. Talman, Variable dimension simplicial algorithm for balanced games. Discussion paper 9025, Center Tilburg University, 1991bGoogle Scholar
  86. 86.
    S. Karamardian, The complementarity problem. Math. Program. 2, 107–129 (1972)MathSciNetMATHGoogle Scholar
  87. 87.
    S. Karamardian, Fixed Points: Algorithms and Applications (Academic, New York, 1977)MATHGoogle Scholar
  88. 88.
    D. Köberl, The solution of nonlinear equations by the computation of fixed points with a modification of the Sandwich method. Computing 25, 175–179 (1980)MathSciNetMATHGoogle Scholar
  89. 89.
    M. Kojima, On the homotopic approach to systems of equations with separable mappings. Math. Program. Study 7, 170–184 (1978a)MathSciNetMATHGoogle Scholar
  90. 90.
    M. Kojima, A modification of Todd’s triangulation J 3. Math. Program. 15, 223–237 (1978b)MathSciNetMATHGoogle Scholar
  91. 91.
    M. Kojima, Studies on piecewise-linear approximations of piecewise-C 1-mappings in fixed points and complementarity theory. Math. Oper. Res. 3, 17–36 (1978c)MathSciNetMATHGoogle Scholar
  92. 92.
    M. Kojima, H. Nishino, N. Arima, A PL homotopy for finding all the roots of a polynomial. Math. Program. 16, 37–62 (1979)MathSciNetMATHGoogle Scholar
  93. 93.
    M. Kojima, R. Saigal, On the number of solutions to a class of complementarity problems. Math. Program. 21, 190–203 (1981)MathSciNetMATHGoogle Scholar
  94. 94.
    M. Kojima, Y. Yamamoto, Variable dimension algorithms: basic theory, interpretation, and extensions of some existing methods. Math. Program. 24, 177–215 (1982)MathSciNetMATHGoogle Scholar
  95. 95.
    M. Kojima, Y. Yamamoto, A unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm. Math. Program. 28, 288–328 (1984)MathSciNetMATHGoogle Scholar
  96. 96.
    J.W.A.M. Kremers, A.J.J. Talman, Solving the nonlinear complementarity problem. Methods Oper. Res. 62, 91–103 (1990)MathSciNetGoogle Scholar
  97. 97.
    H.W. Kuhn, Simplicial approximation of fixed points. Proc. Natl. Acad. Sci. 61, 1238–1242 (1968)MATHGoogle Scholar
  98. 98.
    H.W. Kuhn, Approximate search for fixed points. Comput. Methods Optim. Probl. 2, 199–211 (1969)Google Scholar
  99. 99.
    H.W. Kuhn, A new proof of the fundamental theorem of algebra. Math. Program. Study 1, 148–158 (1974)Google Scholar
  100. 100.
    H.W. Kuhn, Finding roots of polynomials by pivoting, in Fixed Points: Algorithms and Applications, ed. by S. Karamardian (Academic, New York, 1977), pp. 11–39Google Scholar
  101. 101.
    H.W. Kuhn, J.G. MacKinnon, The sandwich method for finding fixed points. J. Optim. Theory Appl. 17, 189–204 (1975)MathSciNetMATHGoogle Scholar
  102. 102.
    H.W. Kuhn, Z. Wang, S. Xu, On the cost of computing roots of polynomials. Math. Program. 28, 156–163 (1984)MathSciNetMATHGoogle Scholar
  103. 103.
    J.C. Lagarias, The computational complexity of simultaneous Diophantine approximation problems. SIAM J. Comput. 14, 196–209 (1985)MathSciNetMATHGoogle Scholar
  104. 104.
    C.W. Lee, Triangulating the d-cube. IBM Thomas J. Watson Research Centre Technical Report, Yorktown Heights, New York, 1984Google Scholar
  105. 105.
    C.E. Lemke, Bimatrix equilibrium points and mathematical programming. Manage. Sci. 11, 681–689 (1965)MathSciNetMATHGoogle Scholar
  106. 106.
    C.E. Lemke, J.T. Howson, Equilibrium points of bimatrix games. SIAM Rev. 12, 413–423 (1964)MathSciNetMATHGoogle Scholar
  107. 107.
    H.J. Luthi, A simplicial approximation of a solution for the nonlinear complementarity problem. Math. Program. 9, 278–293 (1975)MathSciNetGoogle Scholar
  108. 108.
    J.G. Mackinnon, Solving urban general equilibrium problems by fixed point methods, in Mathematical Programming, ed. by S. Robinson, Analysis and Computation of Fixed Points (Academic, New York, 1980), pp. 197–212Google Scholar
  109. 109.
    O.L. Mangasarian, Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976)MathSciNetMATHGoogle Scholar
  110. 110.
    A. Mansur, J. Whalley, A decomposition algorithm for general equilibrium computation with application to international trade models. Econometrica 50, 1547–1557 (1982)MathSciNetMATHGoogle Scholar
  111. 111.
    P.S. Mara, Triangulations for the cube. J. Comb. Theory A 20, 170–177 (1976)MathSciNetMATHGoogle Scholar
  112. 112.
    A. Mas-Colell, The Theory of General Economic Equilibrium. Econometric Society Publication, vol. 9 (Cambridge University Press, Cambridge, MA, 1985)Google Scholar
  113. 113.
    N. Megiddo, On the parametric nonlinear complementarity problem. Math. Program. Study 7, 142–150 (1978)MathSciNetMATHGoogle Scholar
  114. 114.
    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, Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, 1972Google Scholar
  115. 115.
    M. Meyerson A.H. Wright, A new and constructive proof of the Borsuk-Ulam theorem. Proc. Am. Math. Soc. 73, 134–136 (1979)MathSciNetMATHGoogle Scholar
  116. 116.
    K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming. Sigma Series in Applied Mathematics, vol. 3 (Heldermann, Berlin, 1988)MATHGoogle Scholar
  117. 117.
    A.N. Netravali, R. Saigal, Optimal quantizer design using a fixed point algorithm. Bell Syst. Tech. J. 55, 1423–1435 (1976)MathSciNetGoogle Scholar
  118. 118.
    J.M. Ortega, W.C. Rheinboldt, Iterative Solutions of Nonlinear Equations of Several Variables (Academic, New York, 1970)Google Scholar
  119. 119.
    H.O. Peitgen, Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol. 730 (Springer, New York/Berlin, 1979)MATHGoogle Scholar
  120. 120.
    M. Prüfer, H.W. Siegberg, Complementarity pivoting and the Hopf degree theorem. J. Math. Anal. Appl. 84, 133–149 (1981)MathSciNetMATHGoogle Scholar
  121. 121.
    P.M. Reiser, A modified integer labeling for complementarity algorithms. Math. Oper. Res. 6, 129–139 (1981)MathSciNetMATHGoogle Scholar
  122. 122.
    J. Renegar, On the complexity of a piecewise linear algorithm for approximating roots for complex polynomials. Math. Program. 32, 301–318 (1985a)MathSciNetMATHGoogle Scholar
  123. 123.
    J. Renegar, On the cost of approximating all roots of a complex polynomial. Math. Program. 32, 319–336 (1985b)MathSciNetMATHGoogle Scholar
  124. 124.
    J. Renegar, Rudiments of an average case complexity theory for piecewise-linear path following algorithms. Math. Program. 40, 113–163 (1988)MathSciNetMATHGoogle Scholar
  125. 125.
    S. Robinson, Analysis and Computation of Fixed Points (Academic, New York, 1980)MATHGoogle Scholar
  126. 126.
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)MATHGoogle Scholar
  127. 127.
    P.H.M. Ruys, G. van der Laan, Computation of an industrial equilibrium, in The Computation and Modelling of Economic Equilibria, ed. by A.J.J. Talman, G. van Laan. Conributions to Economic Analysis, vol. 167 (North-Holland, Amsterdam, 1987), pp. 205–230Google Scholar
  128. 128.
    D.G. Saari, R. Saigal, Some generic properties of paths generated by fixed points algorithms, in Analysis and Computation of Fixed Points ed. by S. Robinson (Academic, New York, 1980), pp. 57–72 (1980)Google Scholar
  129. 129.
    D.G. Saari, C.P. Simon, Effective price mechanisms. Econometrica 46, 1097–1125 (1978)MathSciNetMATHGoogle Scholar
  130. 130.
    R. Saigal, On paths generated by fixed point algorithms. Math. Oper. Res. 4, 359–380 (1976)MathSciNetGoogle Scholar
  131. 131.
    R. Saigal, Investigations into the efficiency of fixed point algorithms, in Fixed Points: Algorithms and Applications, ed. by S. Karamardian (Academic, New York, 1977a), pp. 203–223Google Scholar
  132. 132.
    R. Saigal, On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting. Math. Oper. Res. 2, 108–124 (1977b)MathSciNetMATHGoogle Scholar
  133. 133.
    R. Saigal, The fixed point approach to nonlinear programming. Math. Program. Study 10, 142–157 (1979a)MathSciNetMATHGoogle Scholar
  134. 134.
    R. Saigal, On piecewise linear approximations to smooth mappings. Math. Oper. Res. 2, 153–161 (1979b)MathSciNetGoogle Scholar
  135. 135.
    R. Saigal, An efficient procedure for traversing large pieces in fixed point algorithms, in Homotopy Methods and Global Convergence, ed. by B.C. Eaves, F.J. Gould, H.-O. Peitgen, M.J. Todd. NATO Conference Series vol. 13 (Plenum, New York, 1983a), pp. 239–248Google Scholar
  136. 136.
    R. Saigal, A homotopy for solving large, sparse and structured fixed point problems. Math. Oper. Res. 8, 557–578 (1983b)MathSciNetMATHGoogle Scholar
  137. 137.
    R. Saigal, Computational complexity of a piecewise linear homotopy algorithm. Math. Program. 28, 164–173 (1984)MathSciNetMATHGoogle Scholar
  138. 138.
    R. Saigal, M.J. Todd, Efficient acceleration techniques for fixed point algorithms. SIAM J. Numer. Anal. 15, 997–1007 (1978)MathSciNetMATHGoogle Scholar
  139. 139.
    R. Saigal, D. Solow, L.A. Wolsey, A comparative study of two algorithms to compute fixed points over unbounded regions, in Proceedings of VII-th Mathematical Programming Symposium, Stanford, 1975Google Scholar
  140. 140.
    J.F. Sallee, A triangulation of the n-cube. Discret. Math. 40, 81–86 (1982)MathSciNetMATHGoogle Scholar
  141. 141.
    J.F. Sallee, The middle-cut triangulations of n-cube. SIAM J. Discret. Math. 5, 407–419 (1984)MathSciNetMATHGoogle Scholar
  142. 142.
    A.A. Samuel, M.J. Todd, An efficient simplicial algorithm for computing a zero of a convex union of smooth functions. Math. Program. 25, 83–108 (1983)MATHGoogle Scholar
  143. 143.
    D. Saupe, On accelerating PL continuation algorithms by predictor-corrector methods. Math. Program. 23, 87–110 (1982)MathSciNetMATHGoogle Scholar
  144. 144.
    H. Scarf, The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15, 1328–1343 (1967a)MathSciNetMATHGoogle Scholar
  145. 145.
    H. Scarf, The core of an N person game. Econometrica 35, 50–69 (1967b)MathSciNetMATHGoogle Scholar
  146. 146.
    H. Scarf, The Computation of Economic Equilibria (Yale University Press, New Haven, 1973)MATHGoogle Scholar
  147. 147.
    S. Shamir, Two triangulations for homotopy fixed point algorithms with an arbitrary refinement factor, in Analysis and Computation of Fixed Points, ed. by S. Robinson (Academic, New York, 1980), pp. 25–56Google Scholar
  148. 148.
    J.B. Shoven, Applying fixed point algorithms to the analysis of tax policies, in Fixed Points: Algorithms and Applications, ed. by S. Karamardian (Academic, New York, 1977), pp. 403–434Google Scholar
  149. 149.
    D. Solow, Comparative computer results of a new complementary pivot algorithm for solving equality and inequality constrained optimization problems. Math. Program. 18, 213–224 (1981a)MathSciNetGoogle Scholar
  150. 150.
    D. Solow, Homeomorphisms of triangulations with applications to computing fixed points. Math. Program. 20, 213–224 (1981b)MathSciNetMATHGoogle Scholar
  151. 151.
    A.J.J. Talman, Variable Dimension Fixed Point Algorithms and Triangulations. Mathematical Centre Tracts, vol. 128 (Mathematisch Centrum, Amsterdam, 1980)MATHGoogle Scholar
  152. 152.
    A.J.J. Talman, L. Van der Heyden, Algorithms for the linear complementarity problem which allow an arbitrary starting point, in Homotopy Methods and Global Convergence, ed. by B.C. Eaves, F.J. Gould, H.-O. Peitgen, M.J. Todd. NATO Conference Series vol. 13 (Plenum, New York, 1983a), pp. 239–248, 267–286Google Scholar
  153. 153.
    A.J.J. Talman, G. van Laan, The Computation and Modelling of Economic Equilibria. Contributions to Economic Analysis, vol. 167 (North-Holland, Amsterdam, 1987)Google Scholar
  154. 154.
    A.J.J. Talman, Y. Yamamoto, A simplicial algorithm for stationary point problems on polytopes. Math. Oper. Res. 14, 383–399 (1989)MathSciNetMATHGoogle Scholar
  155. 155.
    M.J. Todd, A generalized complementary pivoting algorithm. Math. Program. 6, 243–263 (1974)MathSciNetMATHGoogle Scholar
  156. 156.
    M.J. Todd, The Computation of Fixed Points and Applications. Lecture Notes on Economics and Mathematical Systems, vol. 124 (Sringer, Berlin, 1976a)MATHGoogle Scholar
  157. 157.
    M.J. Todd, On triangulations for computing fixed points. Math. Program. 10, 322–346 (1976b)MathSciNetMATHGoogle Scholar
  158. 158.
    M.J. Todd, Orientation in complementary pivot algorithms. Math. Oper. Res. 1, 54–66 (1976c)MathSciNetMATHGoogle Scholar
  159. 159.
    M.J. Todd, Union jack triangulations, in Fixed Points: Algorithms and Applications, ed. by S. Karamardian (Academic, New York, 1977), pp. 315–336Google Scholar
  160. 160.
    M.J. Todd, Improving the convergence of fixed point algorithms. Math. Program. Study 7, 151–179 (1978a)MathSciNetMATHGoogle Scholar
  161. 161.
    M.J. Todd, On the Jacobian of a function at a zero computed by a fixed point algorithm. Math. Oper. Res. 3, 126–132 (1978b)MathSciNetMATHGoogle Scholar
  162. 162.
    M.J. Todd, A quadratically-convergent fixed point algorithm for economic equilibria and linearly constrained optimization. Math. Program. 18, 111–126 (1980a)MathSciNetMATHGoogle Scholar
  163. 163.
    M.J. Todd, Exploiting structure in piecewise linear homotopy algorithms for solving equations. Math. Program. 18, 233–247 (1980b)MathSciNetMATHGoogle Scholar
  164. 164.
    M.J. Todd, On the computational complexity of piecewise linear homotopy algorithms. Math. Program. 24, 216–224 (1982)MathSciNetMATHGoogle Scholar
  165. 165.
    M.J. Todd, J’: a new triangulation of R n. SIAM J. Algebr. Discr. Methods 5, 244–254 (1984)MathSciNetMATHGoogle Scholar
  166. 166.
    M.J. Todd, ‘Fat’ triangulations, or solving certain nonconvex matrix optimization problems. Math. Program. 31, 123–136 (1985)MathSciNetMATHGoogle Scholar
  167. 167.
    H. Tuy, Pivotal methods for computing equilibrium points: unified approach and a new restart algorithm. Math. Program. 16, 210–227 (1979)MATHGoogle Scholar
  168. 168.
    E. van Damme, Stability and Perfection of Nash Equilibria (Springer, Berlin, 1987)MATHGoogle Scholar
  169. 169.
    L. Van der Heyden, A refinement procedure for computing fixed points. Math. Oper. Res. 7, 295–313 (1982)MathSciNetMATHGoogle Scholar
  170. 170.
    A.H. van den Elzen, A.J.J. Talman, A procedure for finding Nash equilibria in bimatrix games. Methods Models Oper. Res. 35, 27–43 (1991)MATHGoogle Scholar
  171. 171.
    G. van der Laan, Simplicial Fixed Point Algorithms. Mathematical Centre Tracts, vol. 129 (Mathematisch Centrum, Amsterdam, 1980)MATHGoogle Scholar
  172. 172.
    G. van der Laan, Simplicial approximation of unemployment equilibria. J. Math. Econ. 9, 83–97 (1982)MATHGoogle Scholar
  173. 173.
    G. van der Laan, The computation of general equilibrium in economies with a block diagonal pattern. Econometrica 53, 659–665 (1985)MathSciNetMATHGoogle Scholar
  174. 174.
    G. van der Laan, L.P. Seelen, Efficiency and implementation of simplicial zero point algorithms. Math. Program. 30, 196–217 (1984)MATHGoogle Scholar
  175. 175.
    G. van der Laan, A.J.J. Talman, A restart algorithm for computing fixed points without an extra dimension. Math. Program. 17, 74–84 (1979a)MATHGoogle Scholar
  176. 176.
    G. van der Laan, A.J.J. Talman, A restart algorithm without an artificial level for computing fixed points on unbounded regions, in Functional Differential Equations and Approximation of Fixed Points, ed. by H.O. Peitgen. Lecture Notes in Mathematics, vol. 730 (Springer, New York/Berlin, 1979b), pp. 247–2586Google Scholar
  177. 177.
    G. van der Laan, A.J.J. Talman, An improvement of fixed point algorithms by using a good triangulation. Math. Program. 18, 274–285 (1980a)MATHGoogle Scholar
  178. 178.
    G. van der Laan, A.J.J. Talman, A new subdivision for computing fixed points with a homotopy algorithm. Math. Program. 19, 78–91 (1980b)MATHGoogle Scholar
  179. 179.
    G. van der Laan, A.J.J. Talman, Variable dimension restart algorithms for approximating fixed points, in Numerical Solution of Highly Nonlinear Problems ed. by W. Forster (North-Holland, Amsterdam, 1980c), pp. 3–36Google Scholar
  180. 180.
    G. van der Laan, A.J.J. Talman, A class of simplicial restart fixed point algorithms without an extra dimension. Math. Program. 20, 33–48 (1981a)MATHGoogle Scholar
  181. 181.
    G. van der Laan, A.J.J. Talman, On the computation of fixed points in the product space of unit simplices and an application to noncooperative N-person games. Math. Oper. Res. 7, 1–13 (1982)MathSciNetMATHGoogle Scholar
  182. 182.
    G. van der Laan, A.J.J. Talman, Interpretation of the variable dimension fixed point algorithm with artificial level. Math. Oper. Res. 8, 86–99 (1983a)MathSciNetMATHGoogle Scholar
  183. 183.
    G. van der Laan, A.J.J. Talman, Note on the path following approach of equilibrium programming. Math. Program. 25, 363–367 (1983b)MATHGoogle Scholar
  184. 184.
    G. van der Laan, A.J.J. Talman, Simplicial algorithms for finding stationary points, a unifying description. J. Optim. Theory Appl. 50, 262–281 (1986)Google Scholar
  185. 185.
    G. van der Laan, A.J.J. Talman, Adjustment processes for finding economic equilibria, in The Computation and Modelling of Economic Equilibria, ed. by A.J.J. Talman, G. van Laan. Conributions to Economic Analysis, vol. 167 (North-Holland, Amsterdam, 1987a), pp. 205–230, 85–124Google Scholar
  186. 186.
    G. van der Laan, A.J.J. Talman, Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds. Math. Program. 38, 1–15 (1987b)MATHGoogle Scholar
  187. 187.
    G. van der Laan, A.J.J. Talman, Adjustment processes for finding economic equilibrium problems on the unit simplex. Econ. Lett. 23, 119–123 (1987c)Google Scholar
  188. 188.
    G. van der Laan, A.J.J. Talman, L. Van der Heyden, Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling. Math. Oper. Res. 12, 377–397 (1987)MathSciNetMATHGoogle Scholar
  189. 189.
    H.R. Varian, Microeconomic Analysis (W.W. Norton, New York, 1984)Google Scholar
  190. 190.
    A.F. Veinott, G.B. Dantzig, Integral extreme points. SIAM Rev. 10, 371–372 (1968)MathSciNetMATHGoogle Scholar
  191. 191.
    R.S. Wilmuth, A computational comparison of fixed point algorithms which used complementary pivoting, in Fixed Points: Algorithms and Applications, ed. by S. Karamardian (Academic, New York, 1977), pp. 249–280 (1977)Google Scholar
  192. 192.
    A.H. Wright, The octahedral algorithm, a new simplicial fixed point algorithm. Math. Program. 21, 47–69 (1981)MATHGoogle Scholar
  193. 193.
    Y. Yamamoto, A new variable dimension algorithm for the fixed point problem. Math. Program. 25, 329–342 (1983)MATHGoogle Scholar
  194. 194.
    Y. Yamamoto, A path following algorithm for stationary point problems. J. OR Soc. Jpn. 30, 181–198 (1987)MATHGoogle Scholar
  195. 195.
    W.I. Zangwill, An eccentric barycentric fixed point algorithm. Math. Oper. Res. 2, 343–359 (1977)MathSciNetMATHGoogle Scholar
  196. 196.
    W.I. Zangwill, C.B. Garcia, Equilibrium programming: the path following approach and dynamics. Math. Program. 21, 262–289 (1981)MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Systems Engineering and Engineering ManagementCity University of Hong KongKowloonHong Kong

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