Skip to main content
SpringerLink
Log in

An ODE-based approach to nonlinearly constrained minimax problems

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We consider the following problem: Choosex 1, ...,x n to

wherem 1,m 2,m 3 are integers with 0≤m 1≤m 2≤m 3, thef i are given real numbers, and theg i are given smooth functions. Constraints of the formg i (x 1, ...,x n )=0 can also be handled without problem. Each iteration of our algorithm involves approximately solving a certain non-linear system of first-order ordinary differential equations to get a search direction for a line search and using a Newton-like approach to correct back into the feasible region when necessary. The algorithm and our Fortran implementation of it will be discussed along with some examples. Our experience to date has been that the program is more robust than any of the library routines we have tried, although it generally requires more computer time. We have found this program to be an extremely useful tool in diverse areas, including polymer rheology, computer vision, and computation of convexity-preserving rational splines.

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

  1. J. Angelos, M. Henry, E. Kaufman Jr., and T. Lenker, Optimal nodes for polynomial interpolation, (presented at theSixth International Symposium on Approximation Theory, College Station, Texas, 1989), inApproximation Theory VI, vol. 1, eds C.K. Chui, L.L. Schumaker, and J.D. Ward (Academic Press, San Diego, 1989) pp. 17–20.

    Google Scholar 

  2. A. Brown and M. Bartholomew-Biggs, ODE vs. SQP methods for constrained optimisation, Technical Report #179, the Hatfield Polytechnic Optimisation Centre (June, 1987).

  3. C. Dunham and C. Zhu, Best approximation with approximations nonlinear in a few variables, Congr. Numer. 87 (1992) (Proc. Manitoba Conf. on Numerical Math. and Computing) pp. 203–220.

    Google Scholar 

  4. R. Fletcher,Practical Methods of Optimization, 2nd ed. (John Wiley and Sons, 1987).

  5. E. Kaufman and G. Taylor, Approximation and interpolation by convexity preserving rational splines, to appear in Constr. Approx.

  6. K. Schittkowski,More Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems no. 282, (Springer-Verlag, New York, 1987).

    Google Scholar 

  7. P. Wolfe, Finding the nearest point in a polytope, Math. Progr. 11 (1976) 128–149.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, 48859, Mount Pleasant, MI, USA

    E. H. Kaufman Jr.

  2. Department of Mathematics, University of Victoria, V8W 2Y2, Victoria, B. C., Canada

    D. J. Leeming

  3. Department of Mathematics, Colorado State University, 80523, Fort Collins, CO, USA

    G. D. Taylor

Authors
  1. E. H. Kaufman Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. J. Leeming
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. D. Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaufman, E.H., Leeming, D.J. & Taylor, G.D. An ODE-based approach to nonlinearly constrained minimax problems. Numer Algor 9, 25–37 (1995). https://doi.org/10.1007/BF02143925

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02143925

Keywords

Search

Navigation