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A subgradient algorithm for certain minimax and minisum problems

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Abstract

We present a subgradient algorithm for minimizing the maximum of a finite collection of functions. It is assumed that each function is the sum of a finite collection of basic convex functions and that the number of different subgradient sets associated with nondifferentiable points of each basic function is finite on any bounded set. Problems belonging to this class include the linear approximation problem and both the minimax and minisum problems of location theory. Convergence of the algorithm to an epsilon-optimal solution is proven and its effectiveness is demonstrated by solving a number of location problems and linear approximation problems.

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References

  1. I. Barrodale and A. Young, “Computational experience in solving linear operator equations using the Chebyshev norm”, in: S.G. Hayes, ed.,Numerical approximation to functions and data (The Athlone Press, London, 1970) pp. 115–142.

    Google Scholar 

  2. D.P. Bertsekas and S. Mitter, “A descent numerical method for optimization problems with nondifferentiable cost functionals”,SIAM Journal on Control 11 (1973) 637–652.

    Google Scholar 

  3. F.H. Clarke, “Generalized gradients and applications”,Transactions of the American Mathematical Society 205 (1975) 247–262.

    Google Scholar 

  4. J. Cullum, W.E. Donath and P. Wolfe, “The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices”,Mathematical Programming Study 3 (1975) 35–55.

    Google Scholar 

  5. P.M. Dearing and R.L. Francis, “A network flow solution to a multifacility minimax location problem involving rectilinear distances”,Transportation Science 8 (1974) 126–141.

    Google Scholar 

  6. V.F. Dem'yanov and V.N. Malozemov,Introduction to minimax (John Wiley, 1974).

  7. J. Elzinga, D.W. Hearn and W.D. Randolph, “Minimax multifacility location with Euclidean distances”,Transportation Science 10 (1976) 321–336.

    Google Scholar 

  8. J.W. Eyster, J.A. White and W.W. Wierwille, “On solving multifacility location problems using a hyperboloid approximation procedure”,A.I.I.E. Transactions 5 (1973) 1–6.

    Google Scholar 

  9. A. Feuer, “An implementable mathematical programming algorithm for admissible fundamental functions”, Ph.D. Dissertation, Department of Mathematics, Columbia University, New York (1974).

    Google Scholar 

  10. E.G. Gilbert, “An iterative procedure for computing the minimum of a quadratic form on a convex set”,SIAM Journal on Control 4 (1966) 61–80.

    Google Scholar 

  11. A.A. Goldstein, “Optimization of Lipschitz continuous functions”,Mathematical Programming 13 (1977) 14–22.

    Google Scholar 

  12. J. Greenstadt, “Variations on variable metric methods”,Mathematics of Computation 24 (1970) 1–22.

    Google Scholar 

  13. D.W. Hearn and T.J. Lowe, “A subgradient procedure for the solution of minimax location problems”,Computers and Industrial Engineering 2 (1978) 17–25.

    Google Scholar 

  14. H.W. Kuhn and R.E. Kuenne, “An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics”,Journal of Regional Science 4 (1962) 21–33.

    Google Scholar 

  15. C. Lemarechal, “An extension of Davidon methods to nondifferentiable functions”,Mathematical Programming Study 3 (1975) 95–109.

    Google Scholar 

  16. D.G. Luenberger,Introduction to linear and nonlinear programming (Addison-Wesley, 1973).

  17. R. Mifflin, “An algorithm for constrained optimization with semismooth functions”,Mathematics of Operations Research 2 (1977) 191–207.

    Google Scholar 

  18. Pshenichnyi,Necessary conditions for an extremum (Marcel Dekker, New York, 1971).

    Google Scholar 

  19. R.T. Rockafellar,Convex analysis (Princeton University Press, N.J., 1970).

    Google Scholar 

  20. P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions”,Mathematical Programming Study 3 (1975) 145–173.

    Google Scholar 

  21. P. Wolfe, “Finding the nearest point in a polytope”,Mathematical Programming 11 (1976) 128–144.

    Google Scholar 

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This research was partially supported by the Army Research Office, Triangle Park, NC, under contract number DAH-CO4-75-G-0150, and by NSF grants ENG 16-24294 and ENG 75-10225.

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Chatelon, J.A., Hearn, D.W. & Lowe, T.J. A subgradient algorithm for certain minimax and minisum problems. Mathematical Programming 15, 130–145 (1978). https://doi.org/10.1007/BF01609012

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  • DOI: https://doi.org/10.1007/BF01609012

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