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Discretization methods for the solution of semi-infinite programming problems

Abstract

In the first part of this paper, we prove the convergence of a class of discretization methods for the solution of nonlinear semi-infinite programming problems, which includes known methods for linear problems as special cases. In the second part, we modify and study this type of algorithms for linear problems and suggest a specific method which requires the solution of a quadratic programming problem at each iteration. With this algorithm, satisfactory results can also be obtained for a number of singular problems. We demonstrate the performance of the algorithm by several numerical examples of multivariate Chebyshev approximation problems.

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References

  1. 1.

    Hettich, R., andZencke, P.,Numerische Methoden der Approximation und Semi-infiniten Optimierung, B. G. Teubner, Stuttgart, Germany, 1982.

  2. 2.

    Hettich, R.,An Implementation of a Discretization Method for Semi-Infinite Programming, Mathematical Programming, Vol. 34, pp. 354–361, 1986.

  3. 3.

    Juergens, U.,Zur Konvergenz Semiinfiniter Mehrfachaustauschalgorithmen, PhD Thesis, Universität Hamburg, 1986.

  4. 4.

    Hettich, R., Editor,Semi-Infinite Programming, Springer-Verlag, Berlin, Germany, 1979.

  5. 5.

    Fiacco, A. V., andKortanek, K. D., Editors,Semi-Infinite Programming and Applications, Springer-Verlag, Berlin, Germany, 1983.

  6. 6.

    Tichatschke, R., andNebeling, V.,A Cutting-Plane Method for Quadratic Semi-Infinite Programming Problems, Optimization, Vol. 19, pp. 803–817, 1988.

  7. 7.

    Reemtsen, R.,Modifications of the First Remez Algorithm, SIAM Journal on Numerical Analysis, Vol. 27, pp. 507–518, 1990.

  8. 8.

    Reemtsen, R.,A Note on the Regularization of Discrete Approximation Problems, Journal of Approximation Theory, Vol. 58, pp. 352–357, 1989.

  9. 9.

    Werner, J.,Optimization Theory and Applications, Friedrich Vieweg und Sohn, Braunschweig, Germany, 1984.

  10. 10.

    Curtis, A. R., andPowell, J. M. D.,Necessary Conditions for a Minimax Approximation, Computer Journal, Vol. 8, pp. 358–361, 1966.

  11. 11.

    Wetterling, W.,Procedure RVSA (Revidierte Simplexmethode A), Preprint, University of Enschede, Enschede, Holland, 1974.

  12. 12.

    Goldfarb, D., andIdnani, A.,A Numerically Stable Dual Method for Solving Strictly Convex Quadratic Programs, Mathematical Programming, Vol. 27, pp. 1–33, 1983.

  13. 13.

    Watson, G. A.,A Multiple Exchange Algorithm for Multivariate Chebyshev Approximation, SIAM Journal on Numerical Analysis, Vol. 12, pp. 46–52, 1975.

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Communicated by E. Polak

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Reemtsen, R. Discretization methods for the solution of semi-infinite programming problems. J Optim Theory Appl 71, 85–103 (1991). https://doi.org/10.1007/BF00940041

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Key Words

  • Nonlinear programming
  • semi-infinite programming
  • discretization of semi-infinite programming problems
  • Chebyshev approximation