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On the convergence of a wide range of trust region methods for unconstrained optimization

التفاصيل البيبلوغرافية
العنوان: On the convergence of a wide range of trust region methods for unconstrained optimization
المؤلفون: Powell, M. J. D.
بيانات النشر: Oxford University Press
سنة النشر: 2010
المجموعة: HighWire Press (Stanford University)
مصطلحات موضوعية: Articles
الوصف: We consider trust region methods for seeking the unconstrained minimum of an objective function F ( ), , when the gradient ( ), , is available. The methods are iterative with 1 being given. The new vector of variables k +1 is derived from a quadratic approximation to F that interpolates F ( k ) and , where k is the iteration number. The second derivative matrix of the quadratic approximation, B k say, can be indefinite, because the approximation is employed only if the vector of variables satisfies , where Δ k is a “trust region radius” that is adjusted automatically. Thus the approximation is useful if is sufficiently large and if ‖ B k ‖ and Δ k are sufficiently small. It is proved under mild assumptions that the condition is achieved after a finite number of iterations, where ε is any given positive constant, and then it is usual to end the calculation. The assumptions include a Lipschitz condition on and also F has to be bounded below. The termination property is established in a single theorem that applies to a wide range of trust region methods that force the sequence F ( k ), k = 1, 2, 3, …, to decrease monotonically. Any choice of each symmetric matrix B k is allowed, provided that ‖ B k ‖ is bounded above by a constant multiple of k .
نوع الوثيقة: text
وصف الملف: text/html
اللغة: English
Relation: http://imajna.oxfordjournals.org/cgi/content/short/30/1/289; http://dx.doi.org/10.1093/imanum/drp021
DOI: 10.1093/imanum/drp021
الاتاحة: http://imajna.oxfordjournals.org/cgi/content/short/30/1/289
https://doi.org/10.1093/imanum/drp021
Rights: Copyright (C) 2010, Institute of Mathematics and its Applications
رقم الانضمام: edsbas.4009E148
قاعدة البيانات: BASE
الوصف
DOI:10.1093/imanum/drp021