326
Views
25
CrossRef citations to date
0
Altmetric
Original Articles

A modified Polak–Ribière–Polyak conjugate gradient algorithm for large-scale optimization problems

, &
Pages 397-413 | Received 01 Jan 2011, Accepted 01 Aug 2012, Published online: 18 Mar 2014

References

  • Ahmed , T. and Storey , D. 1990 . Efficient hybrid conjugate gradient techniques . Journal of Optimization Theory and Applications , 64 : 379 – 394 .
  • Al-Baali , A. 1985 . Descent property and global convergence of the Fletcher–Reeves method with inexact line search . IMA Journal of Numerical Analysis , 5 : 121 – 124 .
  • Andrei , N. 2008 . A hybrid conjugate gradient algorithm for unconstrained optimization as a convex combination of Hestenes–Stiefel and Dai–Yuan . Studies in Informatics and Control , 17 : 55 – 70 .
  • Birgin , E. G. , Martinez , J. M. and Raydan , M. 2000 . Nonmonotone spectral projected gradient methods on convex sets . SIAM Journal on Optimization , 10 : 1196 – 1121 .
  • Bongartz , I. , Conn , A. R. , Gould , N. I. and Toint , P. L. 1995 . CUTE: constrained and unconstrained testing environment . ACM Transactions on Mathmatical Software , 21 : 123 – 160 .
  • Broyden , C. G. 1970 . The convergence of a class of double-rank minimization algorithms: 2. The new algorithm . Journal of the Institute of Mathematics and Its Applications , 6 : 222 – 231 .
  • Broyden , C. G. , Dennis , J. E. and Moré , J. J. 1973 . On the local and superlinear convergence of quasi-Newton methods. . Journal of the Institute of Mathematics and Its Applications , 12 : 223 – 246 .
  • Byrd , R. and Nocedal , J. 1989 . A tool for the analysis of quasi-Newton methods with application to unconstrained minimization . SIAM Journal on Numerical Analysis , 26 : 727 – 739 .
  • Byrd , R. , Nocedal , J. and Yuan , Y. 1987 . Global convergence of a class of quasi-Newton methods on convex problems . SIAM Journal on Numerical Analysis , 24 : 1171 – 1189 .
  • Dai , Y. 1997 . Analysis of conjugate gradient methods . Ph.D. Thesis, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences
  • Dai , Y. 2002 . A nonmonotone conjugate gradient algorithm for unconstrained optimization . Journal of Systems Science and Complexity , 15 : 139 – 145 .
  • Dai , Y. 2003 . Convergence properties of the BFGS algorithm . SIAM Journal on Optimization , 13 : 693 – 701 .
  • Dai , Y. , Han , J. , Sun , D. , Yin , H. and Yuan , Y. 1999 . Convergence properties of nonlinear conjugate gradient methods . SIAM Journal on Optimization , 10 : 345 – 358 .
  • Dai , Y. and Liao , L. Z. 2001 . New conjugacy conditions and related nonlinear conjugate methods . Applied Mathematics and Optimization , 43 : 87 – 101 .
  • Dai , Y. and Yuan , Y. 1998 . Nonlinear Conjugate Gradient Methods , Shanghai , , People’s Republic of China : Shanghai Scientific and Technical Publishers .
  • Dai , Y. and Yuan , Y. 2000 . A nonlinear conjugate gradient with a strong global convergence properties . SIAM Journal on Optimization , 10 : 177 – 182 .
  • Dai , Y. and Yuan , Y. 2001 . An efficient hybrid conjugate gradient method for unconstrained optimization . Annals of Operations Research , 103 : 33 – 47 .
  • Dennis , J. E. and Moré , J. J. 1974 . A characterization of superlinear convergence and its application to quasi-Newton methods . Mathematics of Computation , 28 : 549 – 560 .
  • Dolan , E. D. and Moré , J. J. 2002 . Benchmarking optimization software with performance profiles . Mathematical Programming , 91 : 201 – 213 .
  • Fletcher , R. 1970 . A new approach to variable metric algorithms . Computer Journal , 13 : 317 – 322 .
  • Fletcher , R. 1997 . Practical Methods of Optimization. Vol. I: Unconstrained Optimization, second edition , New York , NY : Wiley .
  • Fletcher , R. and Reeves , C. 1964 . Function minimization by conjugate gradients . The Computer Journal , 7 : 149 – 154 .
  • Gilbert , J. C. and Nocedal , J. 1992 . Global convergence properties of conjugate gradient methods for optimization . SIAM Journal on Optimization , 2 : 21 – 42 .
  • Goldfarb , D. 1970 . A family of variable metric updates derived by variational means . Mathematics of Computation , 24 : 23 – 26 .
  • Gould , N. I. , Orban , D. and Toint , P. L. 2003 . CUTEr and SifDec: a constrained and unconstrained testing environment, revised . ACM Transcations on Mathmatical Software , 29 : 373 – 394 .
  • Grippo , L. , Lampariello , F. and Lucidi , S. 1986 . A nonmonotone line search technique for Newton’s method. . SIAM Journal on Numerical Analysis , 23 : 707 – 716 .
  • Grippo , L. and Lucidi , S. 1997 . A globally convergent version of the Polak–Ribière gradient method . Mathematical Programming , 78 : 375 – 391 .
  • Hager , W. W. 1989 . A derivate-based bracketing scheme for univariate minimization and the conjugate gradient method . Computers and Mathematics with Applications , 18 : 779 – 795 .
  • Hager , W. W. and Zhang , H. 2005 . A new conjugate gradient method with guaranteed descent and an efficient line search . SIAM Journal on Optimization , 16 : 170 – 192 .
  • Hager , W. W. and Zhang , H. 2006a . Algorithm 851: CG − DESCENT, a conjugate gradient method with guaranteed descent . ACM Transactions on Mathematical Software , 32 : 113 – 137 .
  • Hager , W. W. and Zhang , H. 2006b . A survey of nonlinear conjugate gradient methods . Pacific Journal of Optimization , 2 : 35 – 58 .
  • Han , J. Y. and Liu , G. H. 1997 . Global convergence analysis of a new nonmonotone BFGS algorithm on convex objective functions . Computational Optimization and Applications , 7 : 277 – 289 .
  • Hestenes , M. R. and Stiefel , E. 1952 . Method of conjugate gradient for solving linear equations . Journal of Research of the National Bureau of Standards , 49 : 409 – 436 .
  • Hu , Y. F. and Storey , C. 1991 . Global convergence result for conjugate method . Journal Optimization Theory and Applications , 71 : 399 – 405 .
  • Li , D. and Fukushima , M. 2001a . A modified BFGS method and its global convergence in nonconvex minimization . Journal of Computational and Applied Mathematics , 129 : 15 – 35 .
  • Li , D. and Fukushima , M. 2001b . On the global convergence of the BFGS method for nonconvex unconstrained optimization problems . SIAM Journal on Optimization , 11 : 1054 – 1064 .
  • Li , G. , Tang , C. and Wei , Z. 2007 . New conjugacy condition and related new conjugate gradient methods for unconstrained optimization problems . Journal of Computational and Applied Mathematics , 202 : 532 – 539 .
  • Liu , G. H. , Han , J. Y. and Sun , D. F. 1995 . Global convergence analysis of the BFGS algorithm with nonmonotone linesearch . Optimization , 34 : 147 – 159 .
  • Liu , G. H. and Peng , J. M. 1992 . The convergence properties of a nonmonotonic algorithm . Journal of Computational Mathematics , 1 : 65 – 71 .
  • Liu , Y. and Storey , C. 1992 . Effcient generalized conjugate gradient algorithms part 1: theory . Journal of Applied Mathematics and Computing , 69 : 17 – 41 .
  • Polak , E. 1969 . The conjugate gradient method in extreme problems . Computational Mathematics and Mathematical Physics , 9 : 94 – 112 .
  • Polak , E. and Ribière , G. 1969 . Note sur la convergence de directions conjugees . Rev. Francaise informat Recherche Opératinelle , 3 : 35 – 43 .
  • Powell , M.J.D . 1984 . Nonconvex minimization calculations and the conjugate gradient method . Lecture Notes in Mathematics , 1066 : 122 – 141 .
  • Powell , M.J.D . 1986 . Convergence properties of algorithm for nonlinear optimization . SIAM Review , 28 : 487 – 500 .
  • Shanno , D. F. 1970 . Conditioning of quazi-Newton methods for function minimization . Mathematics of Computation , 24 : 647 – 656 .
  • Toint , P. L. 1996 . An assessment of non-monotone line search techniques for unconstrained minimization problem . SIAM Journal on Scientific Computing , 17 : 725 – 739 .
  • Wei , Z. , Li , G. and Qi , L. 2006a . New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems . Applied Mathematics and Computation , 179 : 407 – 430 .
  • Wei , Z. , Li , G. and Qi , L. 2006b . New quasi-Newton methods for unconstrained optimization problems . Applied Mathematics and Computation , 175 : 1156 – 1188 .
  • Wei , Z. , Li , G. and Qi , L. 2008 . Global convergence of the PRP conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems . Mathematics of Computation , 77 : 2173 – 2193 .
  • Wei , Z. , Yao , S. and Liu , L. 2006 . The convergence properties of some new conjugate gradient methods . Applied Mathematics and Computation , 183 : 1341 – 1350 .
  • Wei , Z. , Yu , G. Yuan G. and Lian , Z. 2004 . The superlinear convergence of a modified BFGS-type method for unconstrained optimization . Computational Optimization and Applications , 29 : 315 – 332 .
  • Yu , G.H. 2007 . Nonlinear self-scaling conjugate gradient methods for large-scale optimization problems . Doctoral dissertations, Sun Yat-Sen University, China
  • Yuan , G. L. 2009 . Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems . Optimization Letters , 3 : 11 – 21 .
  • Yuan , G. L. and Lu , X. W. 2009 . A modified PRP conjugate gradient method . Annals of Operations Research , 166 : 73 – 90 .
  • Yuan , G. L. , Lu , X. W. and Wei , Z. X. 2009 . A conjugate gradient method with descent direction for unconstrained optimization . Journal of Computational and Applied Mathematics , 233 : 519 – 530 .
  • Yuan , G. L. and Wei , Z. X. 2008 . The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex objective functions . Acta Mathematica Sinica, English Series , 24 : 35 – 42 .
  • Yuan , G. L. and Wei , Z. X. 2009 . New line search methods for unconstrained optimization . Journal of the Korean Statistical Society , 38 : 29 – 39 .
  • Yuan , G. L. and Wei , Z. X. 2010 . Convergence analysis of a modified BFGS method on convex minimizations . Computational Optimization and Applications , 47 : 237 – 255 .
  • Yuan , Y. X. 1993 . Analysis on the conjugate gradient method . Optimization methods and Software , 2 : 19 – 29 .
  • Zhang , H. C. and Hager , W. W. 2004 . A nonmonotone line search technique and its application to unconstrained optimization . SIAM Journal on Optimization , 14 : 1043 – 1056 .
  • Zhang , L. , Zhou , W. and Li , D. 2006 . A descent modified Polak-Ribière-Polyak conjugate method and its global convergence . IMA Journal on Numerical Analysis , 26 : 629 – 649 .
  • Zhou , J. L. and Tits , A. L. 1993 . Nonmonotone line search for minimax problem . Journal Optimization Theory and Applications , 76 : 455 – 476 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.