Abstract
Many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large [2]. We consider how large these approximations have to be, if they prevent convergence when the objective function is bounded below and continuously differentiable. Thus we obtain a useful convergence result in the case when there is a bound on the second derivative approximations that depends linearly on the iteration number.
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References
R. Fletcher,Practical methds of optimization, Volume 1: Unconstrained optimization (Wiley, Chichester, 1980).
M.J.D. Powell, “Convergence properties of a class of minimization algorithms”, in: O.L. Managasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 2 (Academic Press, New York, 1975).
D.C. Sorensen, “An example concerning quasi-Newton estimation of a sparse Hessian”,SIGNUM Newsletter 16(2) (1981) 8–10.
D.C. Sorensen, “Trust region methods for unconstrained optimization” in: M.J.D. Powell ed.,Nonlinear Optimization 1981 (Academic Press, London, 1982).
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Powell, M.J.D. On the global convergence of trust region algorithms for unconstrained minimization. Mathematical Programming 29, 297–303 (1984). https://doi.org/10.1007/BF02591998
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DOI: https://doi.org/10.1007/BF02591998