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Global and superlinear convergence of an algorithm for one-dimensional minimization of convex functions

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Abstract

This paper studies an algorithm for minimizing a convex function based upon a combination of polyhedral and quadratic approximation. The method was given earlier, but without a good specification for updating the algorithm's curvature matrix. Here, for the case of onedimensional minimization, we provide a specification that insures convergence even in cases where the curvature scalar tends to zero or infinity. Under mild additional assumptions, we show that the convergence is superlinear.

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References

  1. E.W. Cheney and A.A. Goldstein, “Newton's methods for convex programming and Tchebycheff approximation”,Numerische Mathematik 1 (1959) 253–268.

    Google Scholar 

  2. J.E. Kelley, “The cutting plane method for solving convex programs”,Journal of the Society for Industrial and Applied Mathematics 8 (1960) 703–712.

    Google Scholar 

  3. C. Lemarechal, “Nonsmooth optimization and descent methods”, RR 78-4, International Institute for Applied Systems Analysis, Laxenburg, Austria (1978).

    Google Scholar 

  4. R. Mifflin, “A modification and extension of Lemarechal's algorithm for nonsmooth minimization”, in: D. Sorensen and R. Wets, eds.,Nondifferential and variational techniques in optimization, Mathematical Programming Study 17 (1982) 77–90.

    Google Scholar 

  5. R. Mifflin, “A superlinearly convergent algorithm for one-dimensional constrained minimization problems with convex functions”,Mathematics of Operations Research, to appear.

  6. W. Murray and M.L. Overton, “Steplength algorithms for minimizing a class of nondifferentiable functions”,Computing 23 (1979) 309–331.

    Google Scholar 

  7. R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, NJ, 1970).

    Google Scholar 

  8. P. Wolfe, “Sufficient minimization of piecewise-linear univariate functions, in: C. Lemarechal and R. Mifflin, eds.,Nonsmooth optimization (Pergamon Press, London, 1978) 127–130.

    Google Scholar 

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Authors and Affiliations

  1. INRIA, B.P. 105, 78150, Le Chesnay, France

    C. Lemarechal

  2. Department of Pure and Applied Mathematics, Washington State University, 99164, Pullman, WA, USA

    R. Mifflin

Authors
  1. C. Lemarechal
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  2. R. Mifflin
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Lemarechal, C., Mifflin, R. Global and superlinear convergence of an algorithm for one-dimensional minimization of convex functions. Mathematical Programming 24, 241–256 (1982). https://doi.org/10.1007/BF01585109

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  • DOI: https://doi.org/10.1007/BF01585109

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