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On the global convergence of trust region algorithms for unconstrained minimization. (English) Zbl 0569.90069

This paper presents and proves a result on the convergence of trust region algorithms for unconstrained minimization. The result gives a condition on the norms of the approximations to the Hessian matrix which ensures convergence.
Specifically, the problem min F(x), \(x\in R^ n\), is considered. The general trust region algorithm and auxiliary assumptions are defined precisely in the paper. The algorithm involves iteration of the form \(x_{k+1}=x_ k+\delta_ k\) where \(\delta_ k\) is usually chosen to satisfy \(\phi_ k(x_ k+\delta_ k)<F(x_ k)\), where \(\phi_ k\) is the quadratic model: \(\phi_ k(x_ k+\delta)=F(x_ k)+\delta^ Tg_ k+(1/2)\delta^ TB_ k\delta\). Here, \(g_ k=\nabla F(x_ k)\), and \(B_ k\) is some approximation to the Hessian matrix of F.
Loosely, the result is: if \(| B_ k| \leq c_ 1+c_ 2k\), then \(\lim \inf_{k\to \infty}| g_ k| =0\). The result sharpens a previously published result by the author [Nonlin. Program. 2, Proc. Math. Program. Symp., Madison 1974, 1-27 (1975; Zbl 0321.90045)].
Reviewer: B.Kearfott

MSC:

90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
65H10 Numerical computation of solutions to systems of equations

Citations:

Zbl 0321.90045
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References:

[1] R. Fletcher,Practical methds of optimization, Volume 1: Unconstrained optimization (Wiley, Chichester, 1980). · Zbl 0439.93001
[2] 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). · Zbl 0321.90045
[3] D.C. Sorensen, ”An example concerning quasi-Newton estimation of a sparse Hessian”,SIGNUM Newsletter 16(2) (1981) 8–10. · doi:10.1145/1057562.1057564
[4] D.C. Sorensen, ”Trust region methods for unconstrained optimization” in: M.J.D. Powell ed.,Nonlinear Optimization 1981 (Academic Press, London, 1982). · Zbl 0571.90081
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