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A self-scaling memoryless BFGS based conjugate gradient method using multi-step secant condition for unconstrained minimization
Applications of Mathematics, Tome 69 (2024) no. 6, pp. 847-866 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. Based on the self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (SSML-BFGS) method, new conjugate gradient algorithms CG-DESCENT and CGOPT have been proposed by W. Hager, H. Zhang (2005) and Y. Dai, C. Kou (2013), respectively. It is noted that the two conjugate gradient methods perform more efficiently than the SSML-BFGS method. Therefore, C. Kou, Y. Dai (2015) proposed some suitable modifications of the SSML-BFGS method such that the sufficient descent condition holds. For the sake of improvement of modified SSML-BFGS method, in this paper, we present an efficient SSML-BFGS-type three-term conjugate gradient method for solving unconstrained minimization using Ford-Moghrabi secant equation instead of the usual secant equations. The method is shown to be globally convergent under certain assumptions. Numerical results compared with methods using the usual secant equations are reported.
Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. Based on the self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (SSML-BFGS) method, new conjugate gradient algorithms CG-DESCENT and CGOPT have been proposed by W. Hager, H. Zhang (2005) and Y. Dai, C. Kou (2013), respectively. It is noted that the two conjugate gradient methods perform more efficiently than the SSML-BFGS method. Therefore, C. Kou, Y. Dai (2015) proposed some suitable modifications of the SSML-BFGS method such that the sufficient descent condition holds. For the sake of improvement of modified SSML-BFGS method, in this paper, we present an efficient SSML-BFGS-type three-term conjugate gradient method for solving unconstrained minimization using Ford-Moghrabi secant equation instead of the usual secant equations. The method is shown to be globally convergent under certain assumptions. Numerical results compared with methods using the usual secant equations are reported.
DOI : 10.21136/AM.2024.0204-23
Classification : 65K05, 90C06
Keywords: unconstrained optimization; conjugate gradient method; multi-step secant condition; self-scaling; improved Wolfe line search 
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     title = {A self-scaling memoryless {BFGS} based conjugate gradient method using multi-step secant condition for unconstrained minimization},
     journal = {Applications of Mathematics},
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Kim, Yongjin; Jong, Yunchol; Kim, Yong. A self-scaling memoryless BFGS based conjugate gradient method using multi-step secant condition for unconstrained minimization. Applications of Mathematics, Tome 69 (2024) no. 6, pp. 847-866. doi: 10.21136/AM.2024.0204-23

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