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• W2068943497 abstract "A relaxed form of Newton’s method is analyzed for the problem, $min _Omega F$, with $Omega $ a convex subset of a real Banach space X, and $F:X to mathbb{R}^1 $ twice differentiable in the sense of Fréchet. In this iterative scheme, feasible directions are gotten by minimizing local quadratic approximations Q to F, and the relaxation parameters, or step lengths, are obtained from Goldstein’s rule. The local and global convergence theorems established here yield two significant extensions of an earlier theorem of Goldstein for the special case $Omega = X = {text{a}}$ Hilbert space. In one extension, growth rate conditions on the local approximation Q subsume the classical uniform positivity restriction on $F''$; connections are made here with a recently formulated classification scheme for singular and nonsingular extremals. In the second extension, uniform growth rate conditions are replaced by assumptions of the compactness and boundedness type. This development establishes global convergence of the Newton-Goldstein algorithm for a large class of problems with singular critical points." @default.
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• W2068943497 date "1980-11-01" @default.
• W2068943497 modified "2023-09-25" @default.
• W2068943497 title "Newton’s Method and the Goldstein Step-Length Rule for Constrained Minimization Problems" @default.
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• W2068943497 doi "https://doi.org/10.1137/0318050" @default.
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