Two fundamental convergence theorems for nonlinear conjugate gradient methods and their applications |
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| Authors: | Han Jiye Liu Guanghui Sun Defeng Yin Hongxia |
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| Affiliation: | (1) The Academy of Mathematics and Systems Sciences, Institute of Applied Mathematics, 100080 Beijing, China;(2) the Chinese Academy of Sciences, Institute of Applied Mathematics, 100080 Beijing, China;(3) Department of Industrial Engineering and Management Sciences, Northwestern University, 60208 Evanston, IL, USA;(4) School of Mathematics, University of New York Wales, 2052 Sydney, Australia;(5) Hua Luo-keng Institute for Applied Mathematics and Information Science, the Chinese Academy of Sciences, 100039 Beijing, China |
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| Abstract: | Two fundamental convergence theorems are given for nonlinear conjugate gradient methods only under the descent condition. As a result, methods related to the Fletcher-Reeves algorithm still converge for parameters in a slightly wider range, in particular, for a parameter in its upper bound. For methods related to the Polak-Ribiere algorithm, it is shown that some negative values of the conjugate parameter do not prevent convergence. If the objective function is convex, some convergence results hold for the Hestenes-Stiefel algorithm. |
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| Keywords: | Conjugate gradient method descent condition global convergence |
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