Abstract
This paper discusses some properties of trust region algorithms for nonsmooth optimization. The problem is expressed as the minimization of a functionh(f(x), whereh(·) is convex andf is a continuously differentiable mapping from ℝ″ to ℝ‴. Bounds for the second order derivative approximation matrices are discussed. It is shown that Powel’s [7, 8] results hold for nonsmooth optimization.
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References
R. Fletcher, “A model algorithm for composite NDO problems,”Mathematical Programming Studies 17 (1982) 67–76.
R. Fletcher,Practical methods of optimization, Vol. 1, Unconstrained optimization (John Wiley & Sons, New York, 1980).
R. Fletcher,Practical methods of optimization, Vol. 2, Constrained optimization (John Wiley & Sons, New York, 1981).
R. Fletcher, “Second order correction for nondifferentiable optimization”, in: G.A. Watson, ed.,Numerical analysis (Springer-Verlag, Berlin, 1982).
J.J. Móre, “Recent developments in algorithms and software for trust region methods,” in: A. Bachem, M. Grötschel and B. Korte, eds,Mathematical programming, The state of the art (Springer-Verlag, Berlin-Heidelberg, 1983).
M.J.D. Powell, “A new algorithm for unconstrained optimization”, in: J.B. Rosen, O.L. Mangasarian, and K. Ritter, eds.,Nonlinear programming (Academic Press, New York, 1970).
M.J.D. Powell, “Convergence properties of a classs of minimization algorithms”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear programming 2 (Academic Press, New York, 1975).
M.J.D. Powell, “On the global convergence of trust region algorithms for unconstrained minimization”, Report DAMTP 1982/NA7, University of Cambridge.
M.J.D. Powell, “General algorithms for discrete nonlinear approximation calculations”, Report DAMPT 1983/NA2, University of Cambridge.
R.T. Rockaffellar,Convex analysis (Princeton University Press, Princeton, New Jersey, 1970).
R.T. Rockafellar, The theory of subgradent and its application to problems of optimization: Convex and not convex functions (Heldermann-Verlag, West Berlin, 1981).
D.C. Sorensen, “Trust region methods for unconstrained optimization”, in: M.J.D. Powell, ed.,Nonlinear optimization 1981 (Academic Press, London, 1982).
T. Steihaug, “The conjugate gradient method and trust regions in large scale optimization”,SIAM Journal of Numerical Analysis 20 (1983) 626–637.
Ph.L. Toint, “On the superlinear convergence of an algorithm for a sparse minimization problem”,SIAM Journal of Numerical Analysis 16 (1979) 1036–1045.
Y. Yuan, “An example of only linearly convergence of the trust region algorithm for nonsmooth optimization”, Report DAMTP 1983/NA15, University of Cambridge.
Y. Yuan, “On the superlinear convergence of a trust region algorithm for nonsmooth optimization”, Report DAMTP 1983/NA16, University of Cambridge.
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Yuan, Y. Conditions for convergence of trust region algorithms for nonsmooth optimization. Mathematical Programming 31, 220–228 (1985). https://doi.org/10.1007/BF02591750
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DOI: https://doi.org/10.1007/BF02591750