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 Cited in: 140 Publications all top 5 Cited by 239 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 Métivier, Ludovic 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. ...and 139 more Authors all top 5 Cited 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. Series A. Series B 4 SIAM Journal on Scientific Computing 3 International Journal for Numerical Methods in Fluids 3 Inverse Problems 3 SIAM Journal on Imaging Sciences 2 Computers and Fluids 2 Computers & Mathematics with Applications 2 Computer Methods in Applied Mechanics and Engineering 2 Journal of Computational Physics 2 Mathematics of Computation 2 ACM Transactions on Mathematical Software 2 Numerical Functional Analysis and Optimization 2 SIAM Journal on Numerical Analysis 2 Numerical Algorithms 2 Applied Mathematical Modelling 2 SIAM Review 2 Numerical Linear Algebra with Applications 1 IMA Journal of Numerical Analysis 1 Journal of Fluid Mechanics 1 Journal of Statistical Physics 1 Mathematical Methods in the Applied Sciences 1 Applied Mathematics and Computation 1 Applied Mathematics and Optimization 1 Computing 1 Information Sciences 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of the American Statistical Association 1 Journal of the Korean Mathematical Society 1 Mathematics and Computers in Simulation 1 Networks 1 SIAM Journal on Scientific and Statistical Computing 1 Journal of Information & Optimization Sciences 1 Operations Research Letters 1 Applied Numerical Mathematics 1 Optimization 1 Algorithmica 1 Computational Mechanics 1 Mathematical and Computer Modelling 1 Journal of Scientific Computing 1 Science in China. Series A 1 Signal Processing 1 Machine Learning 1 European Journal of Operational Research 1 Journal of Elasticity 1 Journal of Statistical Computation and Simulation 1 Linear Algebra and its Applications 1 Computational Statistics and Data Analysis 1 Boletín de la Sociedad Matemática Mexicana. Third Series 1 Mathematical Problems in Engineering 1 International Journal of Computational Fluid Dynamics 1 Abstract and Applied Analysis 1 Vychislitel’nye Tekhnologii 1 Chaos 1 Journal of Machine Learning Research (JMLR) 1 Journal of Applied Mathematics and Computing 1 Acta Numerica 1 CWI Tracts 1 Advances in Operations Research all top 5 Cited in 23 Fields 94 Operations research, mathematical programming (90-XX) 90 Numerical analysis (65-XX) 20 Calculus of variations and optimal control; optimization (49-XX) 11 Computer science (68-XX) 8 Partial differential equations (35-XX) 8 Mechanics of deformable solids (74-XX) 8 Fluid mechanics (76-XX) 8 Biology and other natural sciences (92-XX) 8 Information and communication theory, circuits (94-XX) 5 Statistics (62-XX) 4 Geophysics (86-XX) 2 Algebraic geometry (14-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Probability theory and stochastic processes (60-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Mathematical logic and foundations (03-XX) 1 Combinatorics (05-XX) 1 Ordinary differential equations (34-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Approximations and expansions (41-XX) 1 Operator theory (47-XX) 1 Geometry (51-XX) 1 Optics, electromagnetic theory (78-XX) Citations by Year