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

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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 En. 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: En × [0, 1]→En of the formH(x, t) = (1 −t)F 0 (x) + t[F(x) − y] where theF 0 in this paper is taken to be linear. Simplicial subdivisionsT k of En × [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.

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Authors and Affiliations

  1. University of Texas, Austin, TX, USA

    A. Charnes

  2. University of Chicago, Chicago, IL, USA

    C. B. Garcia

  3. Rensselaer Polytechnic Institute, Troy, NY, USA

    C. E. Lemke

Authors
  1. A. Charnes
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  2. C. B. Garcia
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  3. C. E. Lemke
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Additional information

The authors respectively: A. Charnes, research partially supported by Proj. No. NR047-021 Contract N00014-75-C-0269; C.B. Garcia, C.E. Lemke, research partially supported by NSF Grant No. MPS75-09443.

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Charnes, A., Garcia, C.B. & Lemke, C.E. Constructive proofs of theorems relating to:F(x) = y, with applications. Mathematical Programming 12, 328–343 (1977). https://doi.org/10.1007/BF01593801

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  • DOI: https://doi.org/10.1007/BF01593801

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