tn swMATH ID: 5140 Software Authors: Nash, Stephen G. Description: Newton-type minimization via the Lanczos method This paper discusses the use of the linear conjugate-gradient method (developed via the Lanczos method) in the solution of large-scale unconstrained minimization problems. It is shown how the equivalent Lanczos characterization of the linear conjugate-gradient method may be exploited to define a modified Newton method which can be applied to problems that do not necessarily have positive-definite Hessian matrices. This derivation also makes it possible to compute a negative-curvature direction at a stationary point. The above mentioned modified Lanczos algorithm requires up to n iterations to compute the search direction, where n denotes the number of variables of the problem. The idea of a truncated Newton method is to terminate the iterations earlier. A preconditioned truncated Newton method is described that defines a search direction which interpolates between the direction defined by a nonlinear conjugate-gradient-type method and a modified Newton direction. Numerical results are given which show the promising performance of truncated Newton methods. Homepage: http://www.netlib.org/opt/tn Programming Languages: C Keywords: linear conjugate-gradient method; Lanczos method; large-scale unconstrained minimization; negative-curvature direction; preconditioned truncated Newton method; Numerical results Related Software: L-BFGS; CUTEr; TNPACK; minpack; LANCELOT; KELLEY; ve08; TRON; GQTPAR; PREQN; CUTE; Algorithm 500; LBFGS-B; CUTEst; HSL-VF05; L-BFGS-B; HSL; mctoolbox; SifDec; SeDuMi Referenced in: 140 Publications all top 5 Referenced by 237 Authors 11 Fasano, Giovanni 11 Roma, Massimo 7 Nash, Stephen G. 6 Navon, Ionel Michael 5 Andrei, Neculai 4 Martínez, José Mario 4 Nocedal, Jorge 4 Plecháč, Petr 4 Wei, Zengxin 3 Caliciotti, Andrea 3 Carstensen, Carsten 3 Deng, Naiyang 3 Di, Zichao 3 Kojima, Masakazu 3 Lucidi, Stefano 3 Schlick, Tamar 3 Spellucci, Peter 3 Yuan, Gonglin 3 Zhang, Jianzhong 2 Brossier, Romain 2 Daescu, Dacian N. 2 de Hoog, Frank Robert 2 Gill, Philip E. 2 Gratton, Serge 2 Grote, Marcus J. 2 Kim, Sunyoung 2 Le Dimet, Francois-Xavier 2 Leyffer, Sven 2 Mercier, Sylvain 2 Morales, José Luis 2 Nahum, Uri 2 O’Leary, Dianne P. 2 Operto, Stéphane 2 Pesenti, Raffaele 2 Tardieu, Nicolas 2 Toint, Philippe L. 2 Vasseur, Xavier 2 Virieux, Jean 2 Wang, Zhi 2 Wild, Stefan M. 2 Xu, Chengxian 2 Xue, Guoliang 2 Zhou, Bojian 2 Zibulevsky, Michael 1 Abdel-Aziz, Mohammedi R. 1 Ahmed, A. I. 1 Akrotirianakis, Ioannis G. 1 Al-Baali, Mehiddin 1 Al-Haik, M. S. 1 Alekseev, Aleksey K. 1 Anderssen, Robert Scott 1 Antil, Harbir 1 Armand, Paul 1 Austin, Anthony P. 1 Balaras, Elias 1 Bank, Randolph E. 1 Bao, Michael 1 Benoist, Joël 1 Bliemer, Michiel C. J. 1 Borthwick, Alistair G. L. 1 Borzì, Alfio 1 Boyd, Stephen Poythress 1 Brás, Carmo P. 1 Bronstein, Alexander M. 1 Bronstein, Michael M. 1 Browne, Edward P. 1 Busseti, Enzo 1 Byers, Ralph 1 Byrd, Richard H. 1 Campos, Juan S. 1 Chakraborty, Suvra Kanti 1 Chang, Haw-Shiuan 1 Chelikowsky, James R. 1 Chen, Chunhui 1 Cheng, Bin 1 Cheung, Ying Kuen K. 1 Chin, Gillian M. 1 Chouzenoux, Emilie 1 Chung, Julianne M. 1 Chung, Peter W. 1 Curtis, Frank E. 1 Cushman, John H. 1 Dai, Yu-Hong 1 Daydé, Michel J. 1 De Leone, Renato 1 De Simone, Valentina 1 Demmel, James Weldon 1 di Serafino, Daniela 1 Di, Zichao Wendy 1 Droegemeier, Kelvin K. 1 Duan, Xiabin 1 Dussault, Jean-Pierre 1 El-Alem, Mahmoud M. 1 Erway, Jennifer B. 1 Fedkiw, Ronald P. 1 Ferris, Michael C. 1 Fliege, Jörg 1 Fogelson, Aaron L. 1 Fosdick, Roger L. 1 Freddi, Francesco ...and 137 more Authors all top 5 Referenced in 64 Serials 17 Computational Optimization and Applications 10 SIAM Journal on Optimization 8 Journal of Optimization Theory and Applications 6 Journal of Computational and Applied Mathematics 6 Optimization Methods & Software 5 Journal of Global Optimization 4 Mathematical Programming. 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