Skip to main content
SpringerLink
Log in

A variable rate refining triangulation

  • Published:
Mathematical Programming Submit manuscript

Abstract

A new variable rate refining subdivisionB 4 for the solution of equations with piecewise linear homotopies and restart homotopies is described. The essential virtue of the subdivisionB 4 is that it offers vast latitude to the user. In particular, the parameters ofB 4 can be set so one restriction ofB 4 is thatJ 3 triangulation and another restriction is the octahedral subdivision. More generally, full flexibility is available in the placement of local focal points, and perhaps most importantly,B 4 permits the new capability of refining coordinates at variable andindependent rates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • I. Bárány, “Subdivisions and triangulations in fixed point algorithms,” International Research Institute for Management Science (Ryleeva, Moscow, U.S.S.R., 1983).

    Google Scholar 

  • M.N. Broadie, “Subdivisions and antiprisms for PL homotopy algorithms,” Systems Optimization Laboratory Technical Report No. 83-14, Department of Operations Research, Stanford University (Stanford CA, 1983).

    Google Scholar 

  • B.C. Eaves, A Course in Triangulations for Solving Equations with Deformations (Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1984).

    MATH  Google Scholar 

  • B.C. Eaves and R. Saigal, “Homotopies for the computation of fixed points on unbounded regions,”Mathematical Programming 3 (1972) 225–237.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Kojima and Y. Yamamoto, “Variable dimension algorithms: Basic theory, interpretation, and extensions of some existing methods,”Mathematical Programming 24 (1982) 177–215.

    Article  MATH  MathSciNet  Google Scholar 

  • G. van der Laan and A.J.J. Talman, “A new subdivision for computing fixed points with a homotopy algorithm,”Mathematical Programming 19 (1980) 78–91.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Mizuno, “A simplicial algorithm for finding all solutions to polynomial systems of equations,” Master Thesis, Department of System Sciences, Tokyo Institute of Technology (Tokyo February., 1981).

    Google Scholar 

  • S. Shamir, “Two new triangulations for homotopy fixed point algorithms with an arbitrary refinement factor,” in: S.M. Robinson, ed.,Analysis and computation of fixed points (Academic Press, New York, 1980), pp. 25–56.

    Google Scholar 

  • M.J. Todd, “On triangulations for computing fixed points,”Mathematical Programming 10 (1976) 322–346.

    Article  MATH  MathSciNet  Google Scholar 

  • M.J. Todd, “Union jack triangulations,” in: S. Karamardian and C.B. Garcia, eds,Fixed points: Algorithms and applications (Academic Press, New York, 1977) pp. 315–336.

    Google Scholar 

  • M.J. Todd, “Improving the convergence of fixed point algorithms,” in: M.K. Balinski and R.W. Cottle, eds.,Mathematical Programming Study 7 (North-Holland, Amsterdam, 1978) pp. 151–169.

    Google Scholar 

  • A.H. Wright, “The octahedral algorithm, a new simplicial fixed point algorithm,”Mathematical Programming 21 (1981) 47–69.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Business, Columbia University, 10027, New York, NY, USA

    Mark N. Broadie

  2. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA

    B. Curtis Eaves

Authors
  1. Mark N. Broadie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Curtis Eaves
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by National Science Foundation Grants DMS 84-04121 and OMS 86-03232.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Broadie, M.N., Eaves, B.C. A variable rate refining triangulation. Mathematical Programming 38, 161–202 (1987). https://doi.org/10.1007/BF02604640

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02604640

Key words

Search

Navigation