HSL-VF05 swMATH ID: 36311 Software Authors: Gould, Nicholas I. M.; Lucidi, Stefano; Roma, Massimo; Toint, Philippe L. Description: Solving the trust-region subproblem using the Lanczos method. The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust-region boundary. We investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trust-region problem within the currently generated Krylov subspace has a very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL-VF05 within the Harwell Subroutine Library. Homepage: https://www.researchgate.net/publication/2263739_Solving_the_Trust-Region_Subproblem_using_the_Lanczos_Method Dependencies: HSL Keywords: convergence acceleration; nonlinear systems; fixed point iteration; conjugate gradients; preconditioning Related Software: GQTPAR; CUTEr; LSTRS; GALAHAD; SifDec; L-BFGS; CUTEst; tn; Ipopt; ARPACK; Regularization tools; LANCELOT; LBFGS-B; LAPACK; Mosek; SNOPT; SparseMatrix; MA57; LSQR; HSL Cited in: 74 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Solving the trust-region subproblem using the Lanczos method. Zbl 1047.90510Gould, Nicholas I. M.; Lucidi, Stefano; Roma, Massimo; Toint, Philippe L. 1999 all top 5 Cited by 150 Authors 5 Gould, Nicholas Ian Mark 5 Shen, Chungen 5 Toint, Philippe L. 5 Zhang, Lei-Hong 4 Salahi, Maziar 4 Taati, Akram 3 Cartis, Coralia 3 Xia, Yong 3 Yang, Weihong 2 Burer, Samuel 2 Carmon, Yair 2 Duchi, John C. 2 Erway, Jennifer B. 2 Griffin, Joshua D. 2 Kılınç Karzan, Fatma 2 Marcia, Roummel F. 2 Nakatsukasa, Yuji 2 Pong, Ting Kei 2 Robinson, Daniel P. 2 Roma, Massimo 2 Wang, Jiulin 2 Yang, Boshi 2 Zhu, Detong 1 Adachi, Satoru 1 Agarwal, Naman 1 Ahookhosh, Masoud 1 Akrotirianakis, Ioannis G. 1 Apostolopoulou, M. S. 1 Beck, Amir 1 Bellis, Cédric 1 Belotti, Pietro 1 Bergou, El Houcine 1 Bianconcini, Tommaso 1 Botsaris, C.-A. E. 1 Boumal, Nicolas 1 Brás, Carmo P. 1 Brossier, Romain 1 Bullins, Brian 1 Burdakov, Oleg P. 1 Chiang, Naiyuan 1 Chieu, Nguyen Huy 1 Curtis, Frank E. 1 Dang, Chuangying 1 D’Apuzzo, Marco 1 De Simone, Valentina 1 di Serafino, Daniela 1 Diouane, Youssef 1 Eldén, Lars 1 Fasano, Giovanni 1 Fatemi, Masoud 1 Feng, Jiang 1 Fourer, Robert 1 Frangioni, Antonio 1 Furini, Fabio 1 Ghaderi, Susan 1 Gleixner, Ambros M. 1 Gong, Lujin 1 Goulart, Paul J. 1 Gould, Nick I. M. 1 Gratton, Serge 1 Hager, William W. 1 Hansen, Per Christian 1 Hazan, Elad 1 Higham, Nicholas J. 1 Ho-Nguyen, Nam 1 Hoffmann, Alexandre 1 Hou, Liusheng 1 Huang, Baohua 1 Jia, Chunxia 1 Jia, Xiaojing 1 Jia, Zhongxiao 1 Júdice, Joaquim João 1 Kirches, Christian 1 Kolvenbach, Philip 1 Koren, Tomer 1 Lass, Oliver 1 Le Van Hien 1 Lenders, Felix 1 Li, Rencang 1 Liang, Xin 1 Liberti, Leo 1 Liuzzi, Giampaolo 1 Lodi, Andrea 1 Lu, Zhaosong 1 Lucidi, Stefano 1 Lukšan, Ladislav 1 Ma, Changfeng 1 Mahdavi-Amiri, Nezam 1 Mahoney, Michael W. 1 Martínez, José Mario 1 Matonoha, Ctirad 1 Métivier, Ludovic 1 Misener, Ruth 1 Mittelmann, Hans Detlef 1 Monteiller, Vadim 1 Monteiro, Renato D. C. 1 Morigi, Serena 1 Morini, Benedetta 1 Nash, Stephen G. 1 Nguyen, Thi Ngan ...and 50 more Authors all top 5 Cited in 30 Serials 12 Computational Optimization and Applications 10 SIAM Journal on Optimization 9 Mathematical Programming. Series A. Series B 4 Optimization Methods & Software 3 BIT 3 Journal of Computational and Applied Mathematics 3 Optimization Letters 3 Mathematical Programming Computation 2 SIAM Journal on Matrix Analysis and Applications 2 Journal of Global Optimization 2 SIAM Review 2 Computational and Applied Mathematics 2 Journal of Applied Mathematics and Computing 1 Inverse Problems 1 Mathematics of Computation 1 ACM Transactions on Mathematical Software 1 Acta Mathematica Vietnamica 1 Applied Mathematics and Computation 1 Numerical Functional Analysis and Optimization 1 Applied Mathematics and Mechanics. (English Edition) 1 Applied Numerical Mathematics 1 Linear Algebra and its Applications 1 Journal of Mathematical Imaging and Vision 1 SIAM Journal on Scientific Computing 1 Abstract and Applied Analysis 1 Optimization and Engineering 1 RAIRO. Operations Research 1 Computational Management Science 1 Frontiers of Mathematics in China 1 Journal of the Operations Research Society of China all top 5 Cited in 12 Fields 68 Operations research, mathematical programming (90-XX) 42 Numerical analysis (65-XX) 9 Calculus of variations and optimal control; optimization (49-XX) 5 Computer science (68-XX) 4 Linear and multilinear algebra; matrix theory (15-XX) 3 Partial differential equations (35-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Information and communication theory, circuits (94-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Probability theory and stochastic processes (60-XX) 1 Geophysics (86-XX) Citations by Year