References
- J. Barzilai and J.M. Borwein . Two-point step size gradient methods . IMA J. Numer. Anal. , 8 : 141 – 148 , 1988 . http://dx.doi.org/10.1093/imanum/8.1.141 .
- J.R. Birge , L. Qi and Z. Wei . Convergence analysis of some methods for minimizing a nonsmooth convex function . J. Optim. Theory Appl. , 97 : 357 – 383 , 1998 . http://dx.doi.org/10.1023/A:1022630801549 .
- J.R. Birge , L. Qi and Z. Wei . A general approach to convergence properties of some methods for nonsmooth convex optimization . Appl. Math. Optim. , 38 : 141 – 158 , 1998 . http://dx.doi.org/10.1007/s002459900086 .
- E.G. Birgin and J.M. Martinez . A spectral conjugate gradient method for unconstrained optimization . Appl. Math. Optim. , 43 : 117 – 128 , 2001 . http://dx.doi.org/10.1007/s00245-001-0003-0 .
- E.G. Birgin , J.M. Martinez and M. Raydan . Nonmonotone spectral projected gradient methods on convex sets . SIAM J. Optim. , 10 : 1196 – 1221 , 2000 . http://dx.doi.org/10.1137/S1052623497330963 .
- J.F. Bonnans , J.C. Gilbert , C. Lemaréchal and C.A. Sagastizàbal . A family of veriable metric proximal methods . Math. Program. , 68 : 15 – 47 , 1995 . http://dx.doi.org/10.1007/BF01585756 .
- Clarke F.H. Optimization and Nonsmooth Analysis Wiley, New York 1983
- Conn , A.R. , Gould , N.I.M. and Toint , P.L. 2000 . Trust-Region Methods , Philadelphia , , USA : SIAM .
- R. Correa and C. Lemaréchal . Convergence of some algorithms for convex minization . Math. Program. , 62 : 261 – 273 , 1993 . http://dx.doi.org/10.1007/BF01585170 .
- Y.H. Dai . Alternate step gradient method . Optimization , 52 : 395 – 415 , 2003 . http://dx.doi.org/10.1080/02331930310001611547 .
- Y.H. Dai and R. Fletcher . Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming . Numer. Math. , 100 : 21 – 47 , 2005 . http://dx.doi.org/10.1007/s00211-004-0569-y .
- Y.H. Dai , W.W. Hager , K. Schittkowski and H. Zhang . The cyclic Barzilai– Borwein method for unconstrained optimization . IMA J. Numer. Anal. , 26 : 604 – 627 , 2006 . http://dx.doi.org/10.1093/imanum/drl006 .
- Y.H. Dai and L.Z. Liao . R-linear convergence of the Barzilai and Borwein gradient method . IMA J. Numer. Anal ., 22 : 1 – 10 , 2002 . http://dx.doi.org/10.1093/imanum/22.1.1 .
- Dai , Y.H. , Liao , L.Z. and Li , D. 2005 . “ An analysis of barzilai–borwein gradient method for unsymmetric linear equations ” . In Optimization and Control with Applications Edited by: Teo , K. , Qi , L. and Yang , X. 183 – 211 . Springer
- Y.H. Dai , J. Yuan and Y.X. Yuan . Modified two-point stepsize gradient method for unconstrained optimization . Comput. Optim. Appl ., 22 : 103 – 109 , 2002 . http://dx.doi.org/10.1023/A:1014838419611 .
- Y.H. Dai and Y. Yuan . Analysis of monotone gradient methods . J. Ind. Manag. Optim , 1 : 181 – 192 , 2005 . http://dx.doi.org/10.3934/jimo.2005.1.181 .
- Y.H. Dai and H. Zhang . Adaptive two-point stepsize gradient algorithm . Numer. Algorithms , 27 : 377 – 385 , 2001 . http://dx.doi.org/10.1023/A:1013844413130 .
- R.J.B. de Sampaio , J.Y. Yuan and W.Y. Sun . Trust region algorithm for nonsmooth optimization . Appl. Math. Comput. , 85 : 109 – 116 , 1997 . http://dx.doi.org/10.1016/S0096-3003(96)00112-9 .
- J.E. Dennis Jr. , S.B. Li and R.A. Tapia . A unified approach to global convergence of trust region methods for nonsmooth optimization . Math. Program. , 68 : 319 – 346 , 1995 . http://dx.doi.org/10.1007/BF01585770 .
- E.D. Dolan and J.J. Moré . Benchmarking optimization software with performance profiles . Math. Program ., 91 : 201 – 213 , 2002 . http://dx.doi.org/10.1007/s101070100263 .
- A. Friedlander , J.M. Martínez and B. Molina . Gradient method with restarts and generalizations . SIAM J. Numer. Anal ., 36 : 275 – 289 , 1999 . http://dx.doi.org/10.1137/S003614299427315X .
- M. Fukushima and L. Qi . A global and superlinearly convergent algorithm for nonsmooth convex minimization . SIAM J. Optim ., 6 : 1106 – 1120 , 1996 . http://dx.doi.org/10.1137/S1052623494278839 .
- Gabriel , S.A. and Pang , J.S. 1994 . “ A trust-region method for constrained nonsmooth equations ” . In Large Scale Optimization-State of the Art , Edited by: Hager , W.W. , Hearn , D.W. and Pardalos , P.M. Dordrecht, Holland : Kluwer Academic Publishers .
- L. Grippo , F. Lampariello and S. Lucidi . A nonmonotone line search technique for Newton's method . SIAM J. Numer. Anal ., 23 : 707 – 716 , 1986 . http://dx.doi.org/10.1137/0723046 .
- L. Grippo and M. Sciandrone . Nonmonotone globalization techniques for the Barzilai–Borwein gradient method . Comput. Optim. Appl ., 23 : 134 – 169 , 2002 . http://dx.doi.org/10.1023/A:1020587701058 .
- J.Y. Han and G.H. Liu . Global convergence analysis of a new nonmonotone BFGS algorithm on convex objective functions . Comput. Optim. Appl ., 7 : 277 – 289 , 1997 . http://dx.doi.org/10.1023/A:1008656711925 .
- L. Han , G. Yu and L. Guan . Multivariate spectral gradient method for unconstrained optimization . Appl. Math. Comput ., 210 : 621 – 630 , 2008 . http://dx.doi.org/10.1016/j.amc.2007.12.054 .
- Hiriart-Urruty , J.B. and Lemmaréchal , C. 1983 . Convex Analysis and Minimization Algorithms II , Berlin, Heidelberg : Spring-Verlag .
- Kiwiel , K.C. 1985 . Methods of Descent for Nondifferentiable Optimization , Berlin , New York : Springer-Verlag .
- K.C. Kiwiel . Proximity control in bundle methods for convex nondifferentiable optimization . Math. Program ., 46 : 105 – 122 , 1990 . http://dx.doi.org/10.1007/BF01585731 .
- K.C. Kiwiel . Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities . Math. Program ., 69 : 89 – 109 , 1995 . http://dx.doi.org/10.1007/BF01585554 .
- Lemaréchal , C. 1980 . Extensions diverses des médthodes de gradient et applications , Paris : Thèse d'Etat .
- C. Lemaréchal . Nondifferentiable optimization . In G.L. Nemhauser , A.H.G. Rinnooy Kan and M.J. Todd , Handbooks in Operations Research and Management Science, vol. 1, Optimization , Amsterdam , 1989 . North-Holland .
- Liu , G.H. and Peng , J.M. 1992 . The convergence properties of a nonmonotonic algorithm . J. Comput. Math , 1 : 65 – 71 .
- L. Lukšan and J. Vlček . Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization . Technical Report No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic , 2000 .
- L. Lukšan and J. Vlček . A bundle-Newton method for nonsmooth unconstrained minimization . Math. Program ., 83 : 373 – 391 , 1998 . http://dx.doi.org/10.1007/BF02680566 .
- B. Martinet . Régularisation d'inéquations variationelles par approximations succcessives . Rev. Francaise d'Aut. Inf. Rech. Opé ., 4 : 154 – 159 , 1970 .
- J.M. Martínez , E.A. Pilotta and M. Raydan . Spectral gradient methods for linearly constrained optimization . J. Optim. Theory Appl ., 125 : 629 – 651 , 2005 . http://dx.doi.org/10.1007/s10957-005-2093-3 .
- H. Qi , L. Qi and D. Sun . Solving KKT system via the trust region and the conjugate gradient method . SIAM J. Optim ., 14 : 439 – 463 , 2004 . http://dx.doi.org/10.1137/S105262340038256X .
- L. Qi . Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems . Math. Oper. Res ., 24 : 440 – 471 , 1999 . http://dx.doi.org/10.1287/moor.24.2.440 .
- L. Qi and J. Sun . A nonsmooth version of Newton's method . Math. Program. , 58 : 353 – 367 , 1993 . http://dx.doi.org/10.1007/BF01581275 .
- M. Raydan . On the Barzilai and Borwein chsoce of steplength for the gradient method . IMA J. Numer. Anal ., 13 : 321 – 326 , 1993 . http://dx.doi.org/10.1093/imanum/13.3.321 .
- M. Raydan . The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem . SIAM J. Optim ., 7 : 26 – 33 , 1997 . http://dx.doi.org/10.1137/S1052623494266365 .
- N. Sagara and M. Fukushima . A trust region method for nonsmooth convex optimization . J. Ind. Manag. Optim ., 1 : 171 – 180 , 2005 . http://dx.doi.org/10.3934/jimo.2005.1.171 .
- H. Schramm and J. Zowe . A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results . SIAM J. Optim ., 2 : 121 – 152 , 1992 . http://dx.doi.org/10.1137/0802008 .
- P.L. Toint . An assessment of non-monotone line search techniques for unconstrained minimization problem . SIAM J. Optim. , 17 : 725 – 739 , 1996 .
- C. Wang , Q. Liu and X. Yang . Convergence properties of nonmonotone spectral projected gradient methods . J. Comput. Appl. Math ., 182 : 51 – 66 , 2005 . http://dx.doi.org/10.1016/j.cam.2004.10.018 .
- Z. Wei and L. Qi . Convergence analysis of a proximal Newton method . Numer. Funct. Anal. Optim. , 17 : 463 – 472 , 1996 . http://dx.doi.org/10.1080/01630569608816705 .
- Z. Wei , L. Qi and J.R. Birge . A new methods for nonsmooth convex optimization . J. Inequal. Appl. , 2 : 157 – 179 , 1998 .
- P. Wolfe . A method of conjugate subgradients for minimizing nondifferentiable convex functions 0. Mathematical Programming Study , 3 : 145 – 173 , 1975 .
- Y. Xiao , Q. Wang and D. Wang . Notes on the Dai–Yuan–Yuan modified spectral gradient method . J. Comput. Appl. Math. , 234 : 2986 – 2992 , 2010 . http://dx.doi.org/10.1016/j.cam.2010.04.012 .
- G. Yu , J. Huang and Y. Zhou . A descent spectral conjugate gradient method for impulse noise removal . Appl. Math. Lett. , 23 : 555 – 560 , 2010 . http://dx.doi.org/10.1016/j.aml.2010.01.010 .
- Z. Yu . Solving bound constrained optimization via a new nonmonotone spectral projected gradient method . Appl. Numer. Math. , 58 : 1340 – 1348 , 2008 . http://dx.doi.org/10.1016/j.apnum.2007.07.007 .
- Z. Yu , J. Lin , J. Sun , Y. Xiao , L. Liu and Z. Li . Spectral gradient projection method for monotone nonlinear equations with convex constraintsl . Appl. Numer. Math ., 59 : 2416 – 2423 , 2009 . http://dx.doi.org/10.1016/j.apnum.2009.04.004 .
- Z. Yu , J. Sun and Y. Qin . A multivariate spectral projected gradient method for bound constrained optimization . J. Comput. Appl. Math ., 235 : 2263 – 2269 , 2011 . http://dx.doi.org/10.1016/j.cam.2010.10.023 .
- H.C. Zhang and W.W. Hager . A nonmonotone line search technique and its application to unconstrained optimization . SIAM J. Optim ., 14 : 1043 – 1056 , 2004 . http://dx.doi.org/10.1137/S1052623403428208 .
- L. Zhang . A new trust region algorithm for nonsmooth convex minimization . Appl. Math. Comput ., 193 : 135 – 142 , 2007 . http://dx.doi.org/10.1016/j.amc.2007.03.059 .
- L. Zhang and W. Zhou . Spectral gradient projection method for solving nonlinear monotone equations . J. Comput. Appl. Math ., 196 : 478 – 484 , 2006 . http://dx.doi.org/10.1016/j.cam.2005.10.002 .
- J.L. Zhou and A.L. Tits . Nonmonotone line search for minimax problem . J. Optim. Theory Appl ., 76 : 455 – 476 , 1993 . http://dx.doi.org/10.1007/BF00939377 .
- Zowe , J. 1985 . “ Nondifferentiable optimization ” . In Computational Mathematical Programming , Edited by: Schittkowski , K. 323 – 356 . Berlin , , Germany : Springer-Verlag .