Mathematical Programming

, Volume 12, Issue 1, pp 328–343 | Cite as

Constructive proofs of theorems relating to:F(x) = y, with applications

  • A. Charnes
  • C. B. Garcia
  • C. E. Lemke


Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space E n . They generalize known theorems.

Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained.

A familiar use is made of homotopies H: E n × [0, 1]→E n of the formH(x, t) = (1 −t)F0(x) + t[F(x) − y] where theF0 in this paper is taken to be linear. Simplicial subdivisionsT k of E n × [0, 1] furnish piecewise linear approximatesG k toH. The basic computation is via the generation of piecewise linear curvesP k which satisfyG k (x, t) = 0. Visualizing a sequence {T k } of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0.

This paper builds upon and continues recent work of C.B. Garcia.


Recent Work Mesh Size Mathematical Method Complementarity Problem Basic Computation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B.C. Eaves, “An odd theorem”,Proceedings of the American Mathematical Society 26 (1970) 509–513.Google Scholar
  2. [2]
    B.C. Eaves, “Homotopies for computation of fixed points”,Mathematical Programming 3 (1972) 1–22.Google Scholar
  3. [3]
    B.C. Eaves, “A short course for solving equations with piecewise linear homotopies”, in:Proceedings of an AMS symposium on nonlinear programming, March 1975, to appear.Google Scholar
  4. [4]
    B.C. Eaves and R. Saigal, “Homotopies for the computation of fixed points on unbounded regions”,Mathematical Programming 3 (1972) 225–237.Google Scholar
  5. [5]
    B.C. Eaves and H.E. Scarf, “The solution of systems of piecewise linear equations”,Mathematics of Operations Research 1 (1976) 1–27.Google Scholar
  6. [6]
    M.L. Fisher, F.J. Gould and J.W. Tolle, “A simplicial approximation algorithm for solving systems of nonlinear equations”, University of Chicago (1974).Google Scholar
  7. [7]
    M.L. Fisher and J.W. Tolle, “A general algorithm for solving the nonlinear complementarity problem”, University of Chicago (June 1975).Google Scholar
  8. [8]
    C.B. Garcia, “Continuation methods for simplicial mappings”, in:Fixed points, algorithms and applications (Academic Press, New York, 1977) pp. 149–163.Google Scholar
  9. [9]
    C.B. Garcia, “A global existence theorem for the equation:Fx = y”, University of Chicago (1975).Google Scholar
  10. [10]
    C.B. Garcia, “Computation of solutions to nonlinear equations under homotopy invariance”,Mathematics of Operations Research, to appear.Google Scholar
  11. [11]
    S. Karamardian, “The complementarity problem”,Mathematical Programming 2 (1972) 107–129.Google Scholar
  12. [12]
    M. Kojima, “A unification of the existence theorems for the nonlinear complementarity problem”,Mathematical Programming 9 (1975) 257–277.Google Scholar
  13. [13]
    O.V. Mangasarian, “Unconstrained methods in nonlinear programming”, in:Proceedings of an AMS symposium on nonlinear programming, March 1975, to appear.Google Scholar
  14. [14]
    N. Megiddo and M. Kojima, “On the existence and uniqueness of solutions in nonlinear complementarity theory”, Res. Rept. No. B. 23, Series B. Department of Information Science, Tokyo Institute of Technology (July 1975).Google Scholar
  15. [15]
    O.H. Merrill, “Applications & extensions of an algorithm that computes fixed points of certain mappings”, Ph.D. Thesis, University of Michigan Ann Arbor, MI (1971).Google Scholar
  16. [16]
    J.J. Moré, “Classes of functions and feasibility conditions in nonlinear complementarity problems”,Mathematical Programming 6 (1974) 327–338.Google Scholar
  17. [17]
    J.M. Ortega and W.C. Rheinboldt,Iterative solutions of nonlinear equations in several variables (Academic Press, New York, 1970).Google Scholar
  18. [18]
    R. Saigal and C. Simon, “Generic properties of the complementarity problem”,Mathematical Programming 4 (1973) 324–335.Google Scholar
  19. [19]
    L.A. Wolsey, “Convergence, simplicial paths and acceleration methods for simplicial approximation algorithms for finding a zero of a system of nonlinear equations”, CORE Discussion Paper No. 7247 (Belgium, 1974).Google Scholar

Copyright information

© The Mathematical Programming Society 1977

Authors and Affiliations

  • A. Charnes
    • 1
  • C. B. Garcia
    • 2
  • C. E. Lemke
    • 3
  1. 1.University of TexasAustinUSA
  2. 2.University of ChicagoChicagoUSA
  3. 3.Rensselaer Polytechnic InstituteTroyUSA

Personalised recommendations