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General models in min-max continous location: Theory and solution techniques

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Abstract

In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of th stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented.

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References

  1. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.

    Google Scholar 

  2. Michelot, C.,The Mathematics of Continuous Location, Studies in Locational Analysis, ISOLDE VI Survey Papers, Vol. 5, pp. 59–83, 1993.

    Google Scholar 

  3. Idrissi, H., Loridan, P., andMichelot, C.,Approximation of Solutions for Location Problems, Journal of Optimization Theory and Applications, Vol. 56, pp. 127–143, 1988.

    Google Scholar 

  4. Ward, J. E., andWendell, R. E.,Using Block Norms for Location Modeling, Operations Research, Vol. 33, pp. 1074–1090, 1985.

    Google Scholar 

  5. Love, R. F., andMorris, J. G.,Modelling Intercity Road Distances by Mathematical Functions, Operational Research Quarterly, Vol. 23, pp. 61–71, 1972.

    Google Scholar 

  6. Berens, W., andKoerling, F. J.,Estimated Road Distances by Mathematical Functions, European Journal of Operational Research, Vol. 21, pp. 54–56, 1972.

    Google Scholar 

  7. Ward, J. E., andWendell, R. E.,A New Norm for Measuring Distance Which Yields Linear Location Problems, Operations Research, Vol. 28, pp. 836–844, 1980.

    Google Scholar 

  8. Plastria, F.,On Destination Optimality in Asymmetric Distance Fermat-Weber Problems, Annals of Operations Research, Vol. 40, pp. 355–369, 1992.

    Google Scholar 

  9. Marti, J. T.,Konvexe Analysis, Birkhauser Verlag, Basel, Switzerland, 1977.

    Google Scholar 

  10. Plastria, F.,Continuous Location Anno 1992: A Progress Report, Studies in Locational Analysis, ISOLDE VI Survey Papers, Vol. 5, pp. 85–127, 1993.

    Google Scholar 

  11. Love, R. F., Morris, J. G., andWesolowsky, G. O.,Facility Location: Models and Methods, Elsevier Science Publishers B.V., North Holland, Amsterdam, Netherlands, 1988.

    Google Scholar 

  12. Rudin, W.,Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill, Auckland, New Zealand, 1976.

    Google Scholar 

  13. Rosenholtz, I.,There is No Differentiable Metric on ℝn, American Mathematical Monthly, Vol. 86, pp. 585–586, 1979.

    Google Scholar 

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

    Google Scholar 

  15. Wendell, R. E., andHurther, A. P., Jr.,Location Theory, Dominance, and Convexity, Operations Research, Vol. 21, pp. 314–320, 1973.

    Google Scholar 

  16. Pshenichnyi, B. N.,Necessary Conditions for an Extremum, Marcel Dekker, New York, New York, 1971.

    Google Scholar 

  17. Frenk, J. B. G., Gromicho, J., andZhang, S.,General Models in Min-Max Planar Location: Checking Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 89, pp. 65–87, 1996.

    Google Scholar 

  18. Avriel, M., Diewert, W. E., Schaible, S., andZang, I.,Generalized Concavity, Plenum Press, New York, New York, 1988.

    Google Scholar 

  19. Komlósi, S.,Some Properties of Nondifferentiable Pseudoconvex Functions, Mathematical Programming, Vol. 26, pp. 232–237, 1983.

    Google Scholar 

  20. Crouzeix, J. P.,Some Differentiability Properties of Quasiconvex Functions on ℝn, Lecture Notes in Control and Information Sciences, Springer Verlag, New York, New York, Vol. 30, pp. 9–20, 1981.

    Google Scholar 

  21. Hiriart-Urruty, J. B., andLemaréchal, C.,Convex Analysis and Minimization Algorithms, I: Fundamentals, Comprehensive Studies in Mathematics, Springer Verlag, Berlin, Germany, Vol. 305, 1993.

    Google Scholar 

  22. Frenk, J. B. G., Gromicho, J., andZhang, S.,A Deep-Cut Ellipsoid Algorithm for Convex Programming: Theory and Applications, Mathematical Programming, Vol. 63, pp. 83–108, 1994.

    Google Scholar 

  23. Shor, N. Z., andGershovich, V. I.,Family of Algorithms for Solving Convex Programming Problems, Cybernetics, Vol. 15, pp. 502–508, 1979.

    Google Scholar 

  24. Ortega, J. M., andRheinboldt, W. C.,Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

    Google Scholar 

  25. Varberg, D. E., andRoberts, A. W.,Convex Functions, Academic Press, New York, New York, 1973.

    Google Scholar 

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

    Google Scholar 

  27. Martinez-Legaz, J. E.,On Lower Subdifferentiable Functions, Trends in Mathematical Optimization, Edited by K. H. Hoffman, J. B. Hiriart-Urruty, C. Lemaréchal, and J. Zowe, Birkhäuser Verlag, Basel, Switzerland, 1988.

    Google Scholar 

  28. Frenk, J. B. G., Gromicho, J., Plastria, F., andZhang, S.,A Deep-Cut Ellipsoid Algorithm and Quasiconvex Programming, Generalized Convexity, Edited by S. Komlósi, T. Rapcsák, and S. Schaible, Springer Verlag, New York, New York, pp. 62–76, 1994.

    Google Scholar 

  29. Bland, R. G., Goldfarb, D., andTodd, M. J.,The Ellipsoid Method: A Survey, Operations Research, Vol. 29, pp. 1039–1091, 1981.

    Google Scholar 

  30. Grötschel, M., Lovász, L., andSchrijver, A.,Geometric Algorithms and Combinatorial Optimization, Springer Verlag, Berlin, Germany, 1988.

    Google Scholar 

  31. Dziuban, S. T., Ecker, J. G., andKupferschmid, M.,Using Deep Cuts in an Ellipsoid Algorithm for Nonlinear Programming, Mathematical Programming Study, Vol. 25, pp. 93–107, 1985.

    Google Scholar 

  32. Goffin, J. L.,Convergence Rates of the Ellipsoid Method on General Convex Functions, Mathematics of Operations Research, Vol. 8, pp. 135–150, 1983.

    Google Scholar 

  33. Shor, N. Z.,Utilization of the Operation of Space Dilation in the Minimization of Convex Functions, Cybernetics, Vol. 6, pp. 7–15, 1970.

    Google Scholar 

  34. Luthi, H. J.,On the Solution of Variational Inequalities by the Ellipsoid Method, Mathematics of Operations Research, Vol. 10, pp. 515–522, 1985.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Econometric Institute, Erasmus University, Rotterdam, Netherlands

    J. B. G. Frenk (Associate Professor of Operations Research) & S. Zhang (Assistant Professor of Operations Research)

  2. DEIO, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal

    J. Gromicho (Assistant Lecturer)

Authors
  1. J. B. G. Frenk
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  2. J. Gromicho
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  3. S. Zhang
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Additional information

Communicated by S. Schaible

The work of the second author was supported by JNICT (Portugal), under Contract BD/631/90-RM, during his stay at Erasmus University in Rotterdam.

The authors would like to thank the anonymous referees for simplifying the proofs in the first part of Section 2 and for their constructive remarks improving the presentation.

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Frenk, J.B.G., Gromicho, J. & Zhang, S. General models in min-max continous location: Theory and solution techniques. J Optim Theory Appl 89, 39–63 (1996). https://doi.org/10.1007/BF02192640

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