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The Newton Bracketing Method for Convex Minimization

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Abstract

An iterative method for the minimization of convex functions f :ℝn → ℝ, called a Newton Bracketing (NB) method, is presented. The NB method proceeds by using Newton iterations to improve upper and lower bounds on the minimum value. The NB method is valid for n = 1, and in some cases for n > 1 (sufficient conditions given here). The NB method is applied to large scale Fermat–Weber location problems.

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References

  1. C. Antoni and A. Dalena, New Trends in Mathematical Programming, Kluwer Acad. Publishers: Boston, 1998, pp. 5–12.

    Google Scholar 

  2. D. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, MA, 1995.

    Google Scholar 

  3. J. Brimberg, “The Fermat—Weber location problem revisited,” Math. Prog., vol. 71, no. 1 (Ser. A), pp. 71–76, 1995.

    Google Scholar 

  4. J. Brimberg, R. Chen, and D. Chen, “Accelerating convergence in the Fermat—Weber location problem,” Oper. Res. Lett. vol. 22, no. 4–5, pp. 151–157, 1998.

    Google Scholar 

  5. J. Brimberg and R.F. Love, “Local convexity results in a generalized Fermat—Weber problem,” Comput. Math. Appl., vol. 37, pp. 87–97, 1999.

    Google Scholar 

  6. R. Chandrasekaran and A. Tamir, “Open questions concerning Weiszfeld's algorithm for the Fermat—Weber location problem,” Math. Prog., vol. 44, no. 3 (ser. A), pp. 293–295, 1989.

    Google Scholar 

  7. R. Chandrasekaran and A. Tamir, “Algebraic optimization: The Fermat—Weber location problem,” Math. Prog., vol. 46, no. 2 (ser. A), pp. 219–224, 1990.

    Google Scholar 

  8. F. Cordellier and J.Ch. Fiorot, “On the Fermat—Weber problem with convex cost functions,” Math. Prog., vol. 14, pp. 295–311, 1978.

    Google Scholar 

  9. Z. Drezner, “The planar two—center and two—median problems,” Transportation Sci., vol. 18, pp. 351–361, 1984.

    Google Scholar 

  10. Z. Drezner, “A note on the Weber location problem,” Ann. Oper. Res., vol. 40, pp. 153–161, 1992.

    Google Scholar 

  11. Z. Drezner, “A note on accelerating the Weiszfeld procedure,” Location Sciences, vol. 3, no. 4, pp. 275–279, 1995.

    Google Scholar 

  12. Z. Drezner and A.J. Goldman, “On the set of optimal points to the Weber problem,” Transportation Sci., vol. 25, pp. 3–8, 1991.

    Google Scholar 

  13. Z. Drezner and D. Simchi-Levi, “Asymptotic behavior of the Weber location problem on the plane,” Ann. Oper. Res., vol. 40, pp. 163–172, 1992.

    Google Scholar 

  14. U. Eckhardt, “Weber's problem andWeiszfeld's algorithm in general spaces,” Math. Prog., vol. 18, no. 2, pp. 186–196, 1980.

    Google Scholar 

  15. J. Elzinga and D.W. Hearn, “On stopping rules for facilities location algorithm,” IIE Transactions, vol. 15, pp. 81–83, 1983.

    Google Scholar 

  16. H. Juel, “On a rational stopping rule for facilities location algorithms,” Naval Res. Logist. Quart., vol. 31, pp. 9–11, 1984.

    Google Scholar 

  17. W. Kaplan and Wei H. Yang, “Duality theorem for a generalized Fermat—Weber problem,” Math. Prog., vol. 76, no. 2 (Ser. A), pp. 285–297, 1997.

    Google Scholar 

  18. I.N. Katz, “Local convergence in Fermat's problem,” Math. Prog., vol. 6, pp. 89–104, 1974.

    Google Scholar 

  19. J. Kiefer, “Sequential minimax search for a maximum,” Proc. Amer. Math. Soc., vol. 4, pp. 502–506, 1953.

    Google Scholar 

  20. H. Kuhn, “A note on Fermat's problem,” Math. Prog., vol. 4, pp. 98–107, 1973.

    Google Scholar 

  21. Y. Levin and A. Ben-Israel, “Directional Newton methods in n variables,” Math. of Computation, vol. 71, pp. 237–250, 2002.

    Google Scholar 

  22. Y. Li, “A Newton acceleration of the Weiszfeld algorithm for minimizing the sum of Euclidean distances,” Comput. Optim. Appl., vol. 10, no. 3, pp. 219–242, 1998.

    Google Scholar 

  23. R. Love and P.D. Dowling, “A generalized bounding method for multifacility location models,” Oper. Res., vol. 37, pp. 89–104, 1989.

    Google Scholar 

  24. R. Love, J. Morris, and G. Wesolowsky, Facilities Location: Models and Methods, North-Holland: Amsterdam, 1988.

    Google Scholar 

  25. R. Love and W. Yeong, “A stopping rule for facilities location algorithms,” AIIE Transactions, vol. 13, pp. 357–362, 1981.

    Google Scholar 

  26. C. Michelot and O. Lefebvre, “A primal-dual algorithm for the Fermat—Weber problem involving mixed gauges,” Math. Prog., vol. 39, pp. 319–335, 1987.

    Google Scholar 

  27. J.G. Morris, “Convergence of the Weiszfeld algorithm for Weber problems using a generalized distance function,” Oper. Res., vol. 29, pp. 37–48, 1981.

    Google Scholar 

  28. J.G. Morris and W.A. Verdini, “Minisum, l p distance location problems solved via a perturbed problem and Weiszfeld's algorithm,” Oper. Res., vol. 27, pp. 1180–1188, 1979.

    Google Scholar 

  29. L.M. Ostresh, Jr., “Convergence and descent in the Fermat location problem,” Transportation Sci., vol. 12, pp. 153–164, 1978.

    Google Scholar 

  30. L.M. Ostresh, Jr., “On the convergence of a class of iterative methods for solving theWeber location problem,” Oper. Res., vol. 26, pp. 597–609, 1978.

    Google Scholar 

  31. M.L. Overton, “A quadratically convergent method for minimizing a sum of Euclidean norms,” Math. Prog., vol. 27, pp. 34–63, 1983.

    Google Scholar 

  32. H. Voß and U. Eckhardt, “Linear convergence of generalized Weiszfeld's method,” Computing, vol. 25, pp. 243–251, 1980.

    Google Scholar 

  33. E. Weiszfeld, “Sur le point per lequel la somme des distances de n points donnés est minimum,” Tohoku Math. J., vol. 43, pp. 355–386, 1937.

    Google Scholar 

  34. S. Wu, “A polynomial time algorithm for solving the Fermat—Weber location problem with mixed norms,” Optimization, vol. 30, no. 3, pp. 227–234, 1994.

    Google Scholar 

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  1. Rutcor–Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Rd, Piscataway, NJ, 08854-8003, USA

    Yuri Levin & Adi Ben-Israel

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  1. Yuri Levin
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  2. Adi Ben-Israel
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Levin, Y., Ben-Israel, A. The Newton Bracketing Method for Convex Minimization. Computational Optimization and Applications 21, 213–229 (2002). https://doi.org/10.1023/A:1013768901780

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