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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
Article

Abstract

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.

Keywords

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.

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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

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