A Hyperbolic Rational Model for Unconstrained Non-Linear Optimization

Abstract

We consider a class of invariant Hyperbolic scaling of a strictly convex quadratic function, to extend the family of the conjugate gradient methods for solving unconstrained minimization problems. An algorithm is derived and evaluated numerically. The results indicate that, in general, the new algorithm is superior to the classical standard CG-algorithm. Keywords: A Hyperbolic Rational Model, Conjugate gradient methods. ‫ا‬ ‫الزائدية‬ ‫النموذج‬‫لنسبية‬ ‫المقيدة‬ ‫غير‬ ‫الخطية‬ ‫غير‬ ‫لالمثلية‬ ‫حسن‬ ‫عباس‬ ‫باسم‬ ‫االسدي‬ ‫حسين‬ ‫نضال‬ ‫ياضيات‬ ‫الر‬ ‫و‬ ‫الحاسبات‬ ‫علوم‬ ‫كلية‬ ، ‫الموصل‬ ‫جامعة‬ ‫البحث:‬ ‫استالم‬ ‫تاريخ‬ 7 / 9 / 2003 ‫البحث:‬ ‫قبول‬ ‫تاريخ‬ 16 / 12 / 2003 ‫الملخص‬ ‫ال‬ ‫النموذج‬ ‫من‬ ‫عمومية‬ ‫اكثر‬ ‫نموذج‬ ‫بناء‬ ‫تم‬ ‫البحث‬ ‫هذا‬ ‫في‬ ‫المشروطة‬ ‫غير‬ ‫لألمثلية‬ ‫بيعي‬ ‫تر‬ ‫ةة‬ ‫بييي‬ ‫التر‬ ‫ةة‬ ‫للدال‬ ‫الثابت‬ ‫المقياس‬ ‫ذات‬ ‫األصناف‬ ‫أحد‬ ‫النوع‬ ‫هذا‬ ‫ويعتبر‬ ‫ائد‬ ‫الز‬ ‫القطع‬ ‫يمثل‬ ‫النموذج‬ ‫وهذا‬ ‫ةروطة‬ ‫المش‬ ‫ةر‬ ‫غي‬ ‫ةة‬ ‫مثلي‬ ‫ا‬ ‫ةائل‬ ‫ما‬ ‫ةل‬ ‫ح‬ ‫ةي‬ ‫ف‬ ‫ةة‬ ‫افق‬ ‫المتر‬ ‫ةات‬ ‫المتها‬ ‫ةر‬ ‫ط‬ ‫وتطوير‬ ‫توسيع‬ ‫لغرض‬ . ‫المحددة‬ ‫الن‬ ‫ان‬ ‫و‬ . ‫ةدديا‬ ‫ع‬ ‫ةت‬ ‫وميم‬ ‫ةدة‬ ‫جدد‬ ‫ةة‬ ‫ارزمي‬ ‫خو‬ ‫اشتقا‬ ‫تم‬ ‫ةة‬ ‫ارزمي‬ ‫اليو‬ ‫أن‬ ‫ةت‬ ‫أأبت‬ ‫ةا‬ ‫أليا‬ ‫ةل‬ ‫التوص‬ ‫ةم‬ ‫ت‬ ‫ةي‬ ‫الت‬ ‫ةائ‬ ‫ت‬ . ‫المهال‬ ‫هذا‬ ‫في‬ ‫فة‬ ‫المعر‬ ‫القياسية‬ ‫ارزمية‬ ‫اليو‬ ‫من‬ ‫كفاءة‬ ‫اكثر‬ ‫هي‬ ‫الهدددة‬ : ‫المفتاحية‬ ‫الكلمات‬ ‫النسبية‬ ‫الزائدية‬ ‫النموذج‬ ‫افق.‬ ‫المتر‬ ‫التدرج‬ ‫ائق‬ ‫طر‬ ,

A New hybrid generalized CG- method for non-linear functions

“…In Table ( 1) we represent comparison between new algorithm with Standard F/R CG-algorithm and Sloboda CG-algorithm. Our numerical results, which are presented in Table (2) confirm that the Hybrid model algorithm is superior to both Standard CG-algorithm and Sloboda CG-algorithm with respect to the total number of NOF and NOI.…”

A Rational Triangle Function as a Model for a Conjugate Gradient Optimization Method.

“…Another model has been developed by Tassopoulos and Storey, [14] as follows: F(q(x) = 1 q(x) + 1/2q(x): 2 > 0 AL-Assady in [3] developed a model as follows :(F(q(x)) = In (q(x)) Al-Bayat, [1] has developed a new rational model which is defined as follows: F(q(x)) = 1 q(x)/1-2 q(x). Also Al-Bayati [4] developed an extended CG algorithm which is based on a general logarithmic model F(q(x) = log(q(x) -1 ) , > 0 And Al-Assady, [2] described there ECG algorithm which is based on the natural log function for the rational q(x) function…”

A New Non Quadratic Model For Unconstrained Non Linear Optimization

“…The implementation of the extended CG method has been performed for general function F(q(x)) of the form of equations (2). The unknown quantities i  were expressed in terms of available quantities of the algorithm .…”

“…And Al-Assady [2] described there ECG algorithm which is based on the natural log function for the rational q(x) function F(q) = log , 2  < 0…”