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The minimization of lower subdifferentiable functions under nonlinear constraints: An all feasible cutting plane algorithm

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Abstract

Nonlinear, possibly nonsmooth, minimization problems are considered with boundedly lower subdifferentiable objective and constraints. An algorithm of the cutting plane type is developed, which has the property that the objective needs to be considered at feasible points only. It generates automatically a nondecreasing sequence of lower bounds converging to the optimal function value, thus admitting a rational rule for stopping the calculations when sufficient precision in the objective value has been obtained. Details are given concerning the efficient implementation of the algorithm. Computational results are reported concerning the algorithm as applied to continuous location problems with distance constraints.

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References

  1. Plastria, F.,Lower Subdifferentiable Functions and Their Minimization by Cutting Planes, Journal of Optimization Theory and Applications, Vol. 46, pp. 37–53, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  2. Cheney, E. W., andGoldstein, A. A.,Newton's Method of Convex Programming and Tchebycheff Approximation, Numerische Mathematik, Vol. 1, pp. 253–268, 1959.

    Article  MathSciNet  Google Scholar 

  3. Kelley, J. E.,The Cutting Plane Method for Solving Convex Programs, SIAM Journal on Applied Mathematics, Vol. 8, pp. 703–712, 1960.

    MathSciNet  Google Scholar 

  4. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton New Jersey, 1970.

    Google Scholar 

  5. Dinkel, J. J., Elliott, W. H., andKochenberger, G. A.,Computational Aspects of Cutting-Plane Algorithms for Geometric Programming Problems, Mathematical Programming, Vol. 13, pp. 200–220, 1977.

    Article  MathSciNet  Google Scholar 

  6. Veinott, A. F. Jr.,The Supporting Hyperplane Method for Unimodal Programming, Operations Research, Vol. 15, pp. 147–152, 1967.

    MATH  MathSciNet  Google Scholar 

  7. Topkis, D. M.,Cutting-Plane Methods without Nested Constrained Sets, Operations Research, Vol. 18, pp. 404–413, 1970; see also the note,ibid., pp. 1216–1220.

    MATH  MathSciNet  Google Scholar 

  8. Eaves, B. C., andZangwill, W. I.,Generalized Cutting Plane Algorithms, SIAM Journal on Control, Vol. 9, pp. 529–542, 1971.

    Article  MathSciNet  Google Scholar 

  9. Plastria, F.,Testing Whether a Cutting Plane May Be Dropped, Revue Belge de Statistique, d'Informatique, et de Recherche Operationnelle, Vol. 22, pp. 11–21, 1982.

    MATH  Google Scholar 

  10. Hansen, P., Peeters, D., andThisse, J. F.,Public Facility Location Models: A Selective Survey, Locational Analysis of Public Facilities, Edited by J. F. Thisse and H. G. Zoller, North-Holland Publishing Company, Amsterdam, Holland, pp. 223–262, 1983.

    Google Scholar 

  11. Plastria, F.,Localization in Single Facility Location, European Journal of Operational Research, Vol. 18, pp. 215–219, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  12. Plastria, F.,Solving General Single Facility Location Problems by Cutting Planes, European Journal of Operational Research, Vol. 29, pp. 98–110, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  13. Plastria, F.,Continuous Location Problems and Cutting Plane Algorithms, Doctoral Thesis, Vrije Universiteit Brussel, Brussels, Belgium, 1983.

    Google Scholar 

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Authors and Affiliations

  1. Centrum voor Statistiek en Operationeel Onderzoek, Vrije Universiteit Brussel, Brussels, Belgium

    F. Plastria (Assistant Lecturer)

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  1. F. Plastria
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Communicated by O. L. Mangasarian

The author thanks the referees for several constructive remarks and for pointing out an error in an earlier version of the proof of Lemma 2.1.

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Plastria, F. The minimization of lower subdifferentiable functions under nonlinear constraints: An all feasible cutting plane algorithm. J Optim Theory Appl 57, 463–484 (1988). https://doi.org/10.1007/BF02346164

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