
-
Previous Article
Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints
- JIMO Home
- This Issue
-
Next Article
Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency
Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization
1. | Department of Applied Mathematics, Graduate School of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan |
2. | Faculty of International Social Sciences, Yokohama National University, 79-4 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan |
3. | Department of Applied Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan |
Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Recently, memoryless quasi-Newton methods based on several kinds of updating formulas were proposed. Since the methods closely related to the conjugate gradient method, the methods are promising. In this paper, we propose a memoryless quasi-Newton method based on the Broyden family with the spectral-scaling secant condition. We focus on the convex and preconvex classes of the Broyden family, and we show that the proposed method satisfies the sufficient descent condition and converges globally. Finally, some numerical experiments are given.
References:
[1] |
M. Al-Baali, Y. Narushima and H. Yabe,
A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Computational Optimization and Applications, 60 (2015), 89-110.
doi: 10.1007/s10589-014-9662-z. |
[2] |
I. Bongart, A. R. Conn, N. I. M. Gould and P. L. Toint,
CUTE: constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160.
doi: 10.1145/200979.201043. |
[3] |
Z. Chen and W. Cheng,
Spectral-scaling quasi-Newton methods with updates from the one parameter of the Broyden family, Journal of Computational and Applied Mathematics, 248 (2013), 88-98.
doi: 10.1016/j.cam.2013.01.012. |
[4] |
W. Y. Cheng and D. H. Li,
Spectral scaling BFGS method, Journal of Optimization Theory and Applications, 146 (2010), 305-319.
doi: 10.1007/s10957-010-9652-y. |
[5] |
Y. H. Dai and C. X. Kou,
A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search, SIAM Journal on Optimization, 23 (2013), 296-320.
doi: 10.1137/100813026. |
[6] |
Y. H. Dai and L. Z. Liao,
New conjugacy conditions and related nonlinear conjugate gradient methods, Applied Mathematics and Optimization, 43 (2001), 87-101.
doi: 10.1007/s002450010019. |
[7] |
E. D. Dolan and J. J. Moré,
Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.
doi: 10.1007/s101070100263. |
[8] |
J. A. Ford and I. A. Moghrabi,
Multi-step quasi-Newton methods for optimization, Journal of Computational and Applied Mathematics, 50 (1994), 305-323.
doi: 10.1016/0377-0427(94)90309-3. |
[9] |
J. C. Gilbert and J. Nocedal,
Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42.
doi: 10.1137/0802003. |
[10] |
N.I.M Gould, D. Orban and P.L. Toint,
CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 373-394.
doi: 10.1145/962437.962439. |
[11] |
W. W. Hager, Hager's web page: https://people.clas.ufl.edu/hager/. Google Scholar |
[12] |
W. W. Hager and H. Zhang,
A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM Journal on Optimization, 16 (2005), 170-192.
doi: 10.1137/030601880. |
[13] |
W. W. Hager and H. Zhang,
A survey of nonlinear conjugate gradient method, Pacific Journal of Optimization, 2 (2006), 35-58.
|
[14] |
W. W. Hager and H. Zhang,
Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent, ACM Transactions on Mathematical Software, 32 (2006), 113-137.
doi: 10.1145/1132973.1132979. |
[15] |
M. R. Hestenes and E. Stiefel,
Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436.
doi: 10.6028/jres.049.044. |
[16] |
S. Hoshino,
A formulation of variable metric methods, IMA Journal of Applied Mathematics, 10 (1972), 394-403.
doi: 10.1093/imamat/10.3.394. |
[17] |
C. X. Kou and Y. H. Dai,
A modified self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno method for unconstrained optimization, Journal of Optimization Theory and Applications, 165 (2015), 209-224.
doi: 10.1007/s10957-014-0528-4. |
[18] |
D. H. Li and M. Fukushima,
A modified BFGS method and its global convergence in nonconvex minimization, Journal of Computational and Applied Mathematics, 129 (2001), 15-35.
doi: 10.1016/S0377-0427(00)00540-9. |
[19] |
F. Modarres, M. A. Hassan and W. J. Leong, Memoryless modified symmetric rank-one method for large-scale unconstrained optimization, American Journal of Applied Sciences, 6 (2009), 2054-2059. Google Scholar |
[20] |
A.U. Moyi and W.J. Leong,
A sufficient descent three-term conjugate gradient method via symmetric rank-one update for large-scale optimization, Optimization, 65 (2016), 121-143.
doi: 10.1080/02331934.2014.994625. |
[21] |
S. Nakayama, Y. Narushima and H. Yabe,
A memoryless symmetric rank-one method with sufficient descent property for unconstrained optimization, Journal of the Operations Research Society of Japan, 61 (2018), 53-70.
doi: 10.15807/jorsj.61.53. |
[22] |
Y. Narushima and H. Yabe,
A survey of sufficient descent conjugate gradient methods for unconstrained optimization, SUT Journal of Mathematics, 50 (2014), 167-203.
|
[23] |
Y. Narushima, H. Yabe and J. A. Ford,
A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM Journal on Optimization, 21 (2011), 212-230.
doi: 10.1137/080743573. |
[24] |
J. Nocedal and S. J. Wright,
Numerical Optimization, 2nd edition, Springer, 2006. |
[25] |
S. S. Oren,
Self-scaling variable metric (SSVM) algorithms, Part Ⅱ: Implementation and experiments, Management Science, 20 (1974), 863-874.
doi: 10.1287/mnsc.20.5.863. |
[26] |
S. S. Oren and D. G. Luenberger,
Self-scaling variable metric (SSVM) algorithms, Part Ⅰ: Criteria and sufficient conditions for scaling a class of algorithms, Management Science, 20 (1974), 845-862.
doi: 10.1287/mnsc.20.5.845. |
[27] |
D. F. Shanno,
Conjugate gradient methods with inexact searches, Mathematics of Operations Research, 3 (1978), 244-256.
doi: 10.1287/moor.3.3.244. |
[28] |
K. Sugiki, Y. Narushima and H. Yabe,
Globally convergent three-term conjugate gradient methods that use secant condition and generate descent search directions for unconstrained optimization, Journal of Optimization Theory and Applications, 153 (2012), 733-757.
doi: 10.1007/s10957-011-9960-x. |
[29] |
L. Sun,
An approach to scaling symmetric rank-one update, Pacific Journal of Optimization, 2 (2006), 105-118.
|
[30] |
W. Sun and Y. Yuan,
Optimization Theory and Methods: Nonlinear Programming, Springer, 2006. |
[31] |
Z. Wei, G. Li and L. Qi,
New quasi-Newton methods for unconstrained optimization problems, Applied Mathematics and Computation, 175 (2006), 1156-1188.
doi: 10.1016/j.amc.2005.08.027. |
[32] |
J. Z. Zhang, N. Y. Deng and L. H. Chen,
New quasi-Newton equation and related methods for unconstrained optimization, Journal of Optimization Theory and Applications, 102 (1999), 147-167.
doi: 10.1023/A:1021898630001. |
[33] |
L. Zhang, W. Zhou and D. H. Li,
A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640.
doi: 10.1093/imanum/drl016. |
[34] |
Y. Zhang and R. P. Tewarson,
Quasi-Newton algorithms with updates from the preconvex part of Broyden's family, IMA Journal of Numerical Analysis, 8 (1988), 487-509.
doi: 10.1093/imanum/8.4.487. |
show all references
References:
[1] |
M. Al-Baali, Y. Narushima and H. Yabe,
A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Computational Optimization and Applications, 60 (2015), 89-110.
doi: 10.1007/s10589-014-9662-z. |
[2] |
I. Bongart, A. R. Conn, N. I. M. Gould and P. L. Toint,
CUTE: constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160.
doi: 10.1145/200979.201043. |
[3] |
Z. Chen and W. Cheng,
Spectral-scaling quasi-Newton methods with updates from the one parameter of the Broyden family, Journal of Computational and Applied Mathematics, 248 (2013), 88-98.
doi: 10.1016/j.cam.2013.01.012. |
[4] |
W. Y. Cheng and D. H. Li,
Spectral scaling BFGS method, Journal of Optimization Theory and Applications, 146 (2010), 305-319.
doi: 10.1007/s10957-010-9652-y. |
[5] |
Y. H. Dai and C. X. Kou,
A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search, SIAM Journal on Optimization, 23 (2013), 296-320.
doi: 10.1137/100813026. |
[6] |
Y. H. Dai and L. Z. Liao,
New conjugacy conditions and related nonlinear conjugate gradient methods, Applied Mathematics and Optimization, 43 (2001), 87-101.
doi: 10.1007/s002450010019. |
[7] |
E. D. Dolan and J. J. Moré,
Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.
doi: 10.1007/s101070100263. |
[8] |
J. A. Ford and I. A. Moghrabi,
Multi-step quasi-Newton methods for optimization, Journal of Computational and Applied Mathematics, 50 (1994), 305-323.
doi: 10.1016/0377-0427(94)90309-3. |
[9] |
J. C. Gilbert and J. Nocedal,
Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42.
doi: 10.1137/0802003. |
[10] |
N.I.M Gould, D. Orban and P.L. Toint,
CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 373-394.
doi: 10.1145/962437.962439. |
[11] |
W. W. Hager, Hager's web page: https://people.clas.ufl.edu/hager/. Google Scholar |
[12] |
W. W. Hager and H. Zhang,
A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM Journal on Optimization, 16 (2005), 170-192.
doi: 10.1137/030601880. |
[13] |
W. W. Hager and H. Zhang,
A survey of nonlinear conjugate gradient method, Pacific Journal of Optimization, 2 (2006), 35-58.
|
[14] |
W. W. Hager and H. Zhang,
Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent, ACM Transactions on Mathematical Software, 32 (2006), 113-137.
doi: 10.1145/1132973.1132979. |
[15] |
M. R. Hestenes and E. Stiefel,
Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409-436.
doi: 10.6028/jres.049.044. |
[16] |
S. Hoshino,
A formulation of variable metric methods, IMA Journal of Applied Mathematics, 10 (1972), 394-403.
doi: 10.1093/imamat/10.3.394. |
[17] |
C. X. Kou and Y. H. Dai,
A modified self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno method for unconstrained optimization, Journal of Optimization Theory and Applications, 165 (2015), 209-224.
doi: 10.1007/s10957-014-0528-4. |
[18] |
D. H. Li and M. Fukushima,
A modified BFGS method and its global convergence in nonconvex minimization, Journal of Computational and Applied Mathematics, 129 (2001), 15-35.
doi: 10.1016/S0377-0427(00)00540-9. |
[19] |
F. Modarres, M. A. Hassan and W. J. Leong, Memoryless modified symmetric rank-one method for large-scale unconstrained optimization, American Journal of Applied Sciences, 6 (2009), 2054-2059. Google Scholar |
[20] |
A.U. Moyi and W.J. Leong,
A sufficient descent three-term conjugate gradient method via symmetric rank-one update for large-scale optimization, Optimization, 65 (2016), 121-143.
doi: 10.1080/02331934.2014.994625. |
[21] |
S. Nakayama, Y. Narushima and H. Yabe,
A memoryless symmetric rank-one method with sufficient descent property for unconstrained optimization, Journal of the Operations Research Society of Japan, 61 (2018), 53-70.
doi: 10.15807/jorsj.61.53. |
[22] |
Y. Narushima and H. Yabe,
A survey of sufficient descent conjugate gradient methods for unconstrained optimization, SUT Journal of Mathematics, 50 (2014), 167-203.
|
[23] |
Y. Narushima, H. Yabe and J. A. Ford,
A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM Journal on Optimization, 21 (2011), 212-230.
doi: 10.1137/080743573. |
[24] |
J. Nocedal and S. J. Wright,
Numerical Optimization, 2nd edition, Springer, 2006. |
[25] |
S. S. Oren,
Self-scaling variable metric (SSVM) algorithms, Part Ⅱ: Implementation and experiments, Management Science, 20 (1974), 863-874.
doi: 10.1287/mnsc.20.5.863. |
[26] |
S. S. Oren and D. G. Luenberger,
Self-scaling variable metric (SSVM) algorithms, Part Ⅰ: Criteria and sufficient conditions for scaling a class of algorithms, Management Science, 20 (1974), 845-862.
doi: 10.1287/mnsc.20.5.845. |
[27] |
D. F. Shanno,
Conjugate gradient methods with inexact searches, Mathematics of Operations Research, 3 (1978), 244-256.
doi: 10.1287/moor.3.3.244. |
[28] |
K. Sugiki, Y. Narushima and H. Yabe,
Globally convergent three-term conjugate gradient methods that use secant condition and generate descent search directions for unconstrained optimization, Journal of Optimization Theory and Applications, 153 (2012), 733-757.
doi: 10.1007/s10957-011-9960-x. |
[29] |
L. Sun,
An approach to scaling symmetric rank-one update, Pacific Journal of Optimization, 2 (2006), 105-118.
|
[30] |
W. Sun and Y. Yuan,
Optimization Theory and Methods: Nonlinear Programming, Springer, 2006. |
[31] |
Z. Wei, G. Li and L. Qi,
New quasi-Newton methods for unconstrained optimization problems, Applied Mathematics and Computation, 175 (2006), 1156-1188.
doi: 10.1016/j.amc.2005.08.027. |
[32] |
J. Z. Zhang, N. Y. Deng and L. H. Chen,
New quasi-Newton equation and related methods for unconstrained optimization, Journal of Optimization Theory and Applications, 102 (1999), 147-167.
doi: 10.1023/A:1021898630001. |
[33] |
L. Zhang, W. Zhou and D. H. Li,
A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640.
doi: 10.1093/imanum/drl016. |
[34] |
Y. Zhang and R. P. Tewarson,
Quasi-Newton algorithms with updates from the preconvex part of Broyden's family, IMA Journal of Numerical Analysis, 8 (1988), 487-509.
doi: 10.1093/imanum/8.4.487. |
name | n | name | n | name | n | name | n |
AKIVA | 2 | DIXMAANC | 3000 | HEART8LS | 8 | PENALTY1 | 1000 |
ALLINITU | 4 | DIXMAAND | 3000 | HELIX | 3 | PENALTY2 | 200 |
ARGLINA | 200 | DIXMAANE | 3000 | HIELOW | 3 | PENALTY3 | 200 |
ARGLINB | 200 | DIXMAANF | 3000 | HILBERTA | 2 | POWELLSG | 5000 |
ARWHEAD | 5000 | DIXMAANG | 3000 | HILBERTB | 10 | POWER | 10000 |
BARD | 3 | DIXMAANH | 3000 | HIMMELBB | 2 | QUARTC | 5000 |
BDQRTIC | 5000 | DIXMAANI | 3000 | HIMMELBF | 4 | ROSENBR | 2 |
BEALE | 2 | DIXMAANJ | 3000 | HIMMELBG | 2 | S308 | 2 |
BIGGS6 | 6 | DIXMAANK | 3000 | HIMMELBH | 2 | SCHMVETT | 5000 |
BOX3 | 3 | DIXMAANL | 3000 | HUMPS | 2 | SENSORS | 100 |
BOX | 10000 | DIXON3DQ | 10000 | JENSMP | 2 | SINEVAL | 2 |
BRKMCC | 2 | DJTL | 2 | KOWOSB | 4 | SINQUAD | 5000 |
BROWNAL | 200 | DQDRTIC | 5000 | LIARWHD | 5000 | SISSER | 2 |
BROWNBS | 2 | DQRTIC | 5000 | LOGHAIRY | 2 | SNAIL | 2 |
BROWNDEN | 4 | EDENSCH | 2000 | MANCINO | 100 | SPARSINE | 5000 |
BROYDN7D | 5000 | EG2 | 1000 | MARATOSB | 2 | SPARSQUR | 10000 |
BRYBND | 5000 | ENGVAL1 | 5000 | MEXHAT | 2 | SPMSRTLS | 4999 |
CHAINWOO | 4000 | ENGVAL2 | 3 | MOREBV | 5000 | SROSENBR | 5000 |
CHNROSNB | 50 | ERRINROS | 50 | MSQRTALS | 1024 | STRATEC | 10 |
CLIFF | 2 | EXPFIT | 2 | MSQRTBLS | 1024 | TESTQUAD | 5000 |
COSINE | 10000 | EXTROSNB | 1000 | NONCVXU2 | 5000 | TOINTGOR | 50 |
CRAGGLVY | 5000 | FLETCBV2 | 5000 | NONDIA | 5000 | TOINTGSS | 5000 |
CUBE | 2 | FLETCHCR | 1000 | NONDQUAR | 5000 | TOINTPSP | 50 |
CURLY10 | 10000 | FMINSRF2 | 5625 | OSBORNEA | 5 | TOINTQOR | 50 |
CURLY20 | 10000 | FMINSURF | 5625 | OSBORNEB | 11 | TQUARTIC | 5000 |
CURLY30 | 10000 | FREUROTH | 5000 | OSCIPATH | 10 | TRIDIA | 5000 |
DECONVU | 63 | GENHUMPS | 5000 | PALMER1C | 8 | VARDIM | 200 |
DENSCHNA | 2 | GENROSE | 500 | PALMER1D | 7 | VAREIGVL | 50 |
DENSCHNB | 2 | GROWTHLS | 3 | PALMER2C | 8 | VIBRBEAM | 8 |
DENSCHNC | 2 | GULF | 3 | PALMER3C | 8 | WATSON | 12 |
DENSCHND | 3 | HAIRY | 2 | PALMER4C | 8 | WOODS | 4000 |
DENSCHNE | 3 | HATFLDD | 3 | PALMER5C | 6 | YFITU | 3 |
DENSCHNF | 2 | HATFLDE | 3 | PALMER6C | 8 | ZANGWIL2 | 2 |
DIXMAANA | 3000 | HATFLDFL | 3 | PALMER7C | 8 | ||
DIXMAANB | 3000 | HEART6LS | 6 | PALMER8C | 8 |
name | n | name | n | name | n | name | n |
AKIVA | 2 | DIXMAANC | 3000 | HEART8LS | 8 | PENALTY1 | 1000 |
ALLINITU | 4 | DIXMAAND | 3000 | HELIX | 3 | PENALTY2 | 200 |
ARGLINA | 200 | DIXMAANE | 3000 | HIELOW | 3 | PENALTY3 | 200 |
ARGLINB | 200 | DIXMAANF | 3000 | HILBERTA | 2 | POWELLSG | 5000 |
ARWHEAD | 5000 | DIXMAANG | 3000 | HILBERTB | 10 | POWER | 10000 |
BARD | 3 | DIXMAANH | 3000 | HIMMELBB | 2 | QUARTC | 5000 |
BDQRTIC | 5000 | DIXMAANI | 3000 | HIMMELBF | 4 | ROSENBR | 2 |
BEALE | 2 | DIXMAANJ | 3000 | HIMMELBG | 2 | S308 | 2 |
BIGGS6 | 6 | DIXMAANK | 3000 | HIMMELBH | 2 | SCHMVETT | 5000 |
BOX3 | 3 | DIXMAANL | 3000 | HUMPS | 2 | SENSORS | 100 |
BOX | 10000 | DIXON3DQ | 10000 | JENSMP | 2 | SINEVAL | 2 |
BRKMCC | 2 | DJTL | 2 | KOWOSB | 4 | SINQUAD | 5000 |
BROWNAL | 200 | DQDRTIC | 5000 | LIARWHD | 5000 | SISSER | 2 |
BROWNBS | 2 | DQRTIC | 5000 | LOGHAIRY | 2 | SNAIL | 2 |
BROWNDEN | 4 | EDENSCH | 2000 | MANCINO | 100 | SPARSINE | 5000 |
BROYDN7D | 5000 | EG2 | 1000 | MARATOSB | 2 | SPARSQUR | 10000 |
BRYBND | 5000 | ENGVAL1 | 5000 | MEXHAT | 2 | SPMSRTLS | 4999 |
CHAINWOO | 4000 | ENGVAL2 | 3 | MOREBV | 5000 | SROSENBR | 5000 |
CHNROSNB | 50 | ERRINROS | 50 | MSQRTALS | 1024 | STRATEC | 10 |
CLIFF | 2 | EXPFIT | 2 | MSQRTBLS | 1024 | TESTQUAD | 5000 |
COSINE | 10000 | EXTROSNB | 1000 | NONCVXU2 | 5000 | TOINTGOR | 50 |
CRAGGLVY | 5000 | FLETCBV2 | 5000 | NONDIA | 5000 | TOINTGSS | 5000 |
CUBE | 2 | FLETCHCR | 1000 | NONDQUAR | 5000 | TOINTPSP | 50 |
CURLY10 | 10000 | FMINSRF2 | 5625 | OSBORNEA | 5 | TOINTQOR | 50 |
CURLY20 | 10000 | FMINSURF | 5625 | OSBORNEB | 11 | TQUARTIC | 5000 |
CURLY30 | 10000 | FREUROTH | 5000 | OSCIPATH | 10 | TRIDIA | 5000 |
DECONVU | 63 | GENHUMPS | 5000 | PALMER1C | 8 | VARDIM | 200 |
DENSCHNA | 2 | GENROSE | 500 | PALMER1D | 7 | VAREIGVL | 50 |
DENSCHNB | 2 | GROWTHLS | 3 | PALMER2C | 8 | VIBRBEAM | 8 |
DENSCHNC | 2 | GULF | 3 | PALMER3C | 8 | WATSON | 12 |
DENSCHND | 3 | HAIRY | 2 | PALMER4C | 8 | WOODS | 4000 |
DENSCHNE | 3 | HATFLDD | 3 | PALMER5C | 6 | YFITU | 3 |
DENSCHNF | 2 | HATFLDE | 3 | PALMER6C | 8 | ZANGWIL2 | 2 |
DIXMAANA | 3000 | HATFLDFL | 3 | PALMER7C | 8 | ||
DIXMAANB | 3000 | HEART6LS | 6 | PALMER8C | 8 |
Method number | Class | Global convergence | |
1 (DFP) | 0 | convex | not established |
2 | 0.25 | convex | established |
3 | 0.5 | convex | established |
4 | 0.75 | convex | established |
5 (BFGS) | 1 | convex | established |
6 | 1.25 | preconvex | established |
7 | 1.5 | preconvex | established |
8 | 1.75 | preconvex | established |
9 | 2 | preconvex | not established |
Method number | Class | Global convergence | |
1 (DFP) | 0 | convex | not established |
2 | 0.25 | convex | established |
3 | 0.5 | convex | established |
4 | 0.75 | convex | established |
5 (BFGS) | 1 | convex | established |
6 | 1.25 | preconvex | established |
7 | 1.5 | preconvex | established |
8 | 1.75 | preconvex | established |
9 | 2 | preconvex | not established |
Method number | Class | Global convergence | |
(BFGS) | 1 | convex | established |
10 | 0.8 | convex | established |
11 | 0.85 | convex | established |
12 | 0.9 | convex | established |
13 | 0.95 | convex | established |
14 | 1.05 | preconvex | established |
15 | 1.1 | preconvex | established |
16 | 1.15 | preconvex | established |
17 | 1.2 | preconvex | established |
Method number | Class | Global convergence | |
(BFGS) | 1 | convex | established |
10 | 0.8 | convex | established |
11 | 0.85 | convex | established |
12 | 0.9 | convex | established |
13 | 0.95 | convex | established |
14 | 1.05 | preconvex | established |
15 | 1.1 | preconvex | established |
16 | 1.15 | preconvex | established |
17 | 1.2 | preconvex | established |
Method number | Class | Global convergence | |
5 (BFGS) | 1 | convex | established |
18 (Hoshino) | convex | established | |
19 | preconvex | established | |
20 | unknown | established | |
21 | preconvex | established |
Method number | Class | Global convergence | |
5 (BFGS) | 1 | convex | established |
18 (Hoshino) | convex | established | |
19 | preconvex | established | |
20 | unknown | established | |
21 | preconvex | established |
Method number | Global convergence | ||
5 (BFGS) | (54) | 1 | established |
19 | (54) | established | |
21 | (54) | established | |
22 (BFGS) | (56) | 1 | not established |
23 | (56) | not established | |
24 | (56) | not established | |
25 (CG_DESCENT) [11,12,14] | established |
Method number | Global convergence | ||
5 (BFGS) | (54) | 1 | established |
19 | (54) | established | |
21 | (54) | established | |
22 (BFGS) | (56) | 1 | not established |
23 | (56) | not established | |
24 | (56) | not established | |
25 (CG_DESCENT) [11,12,14] | established |
[1] |
Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149 |
[2] |
Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe. Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (3) : 595-619. doi: 10.3934/jimo.2013.9.595 |
[3] |
Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237 |
[4] |
Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61 |
[5] |
Yan Gu, Nobuo Yamashita. A proximal ADMM with the Broyden family for convex optimization problems. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020091 |
[6] |
Basim A. Hassan. A new type of quasi-newton updating formulas based on the new quasi-newton equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 227-235. doi: 10.3934/naco.2019049 |
[7] |
Gaohang Yu, Lutai Guan, Guoyin Li. Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. Journal of Industrial & Management Optimization, 2008, 4 (3) : 565-579. doi: 10.3934/jimo.2008.4.565 |
[8] |
Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007 |
[9] |
Saman Babaie–Kafaki, Reza Ghanbari. A class of descent four–term extension of the Dai–Liao conjugate gradient method based on the scaled memoryless BFGS update. Journal of Industrial & Management Optimization, 2017, 13 (2) : 649-658. doi: 10.3934/jimo.2016038 |
[10] |
Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919 |
[11] |
Lijuan Zhao, Wenyu Sun. Nonmonotone retrospective conic trust region method for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 309-325. doi: 10.3934/naco.2013.3.309 |
[12] |
Guanghui Zhou, Qin Ni, Meilan Zeng. A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 595-608. doi: 10.3934/jimo.2016034 |
[13] |
Shenggui Zhang. A sufficient condition of Euclidean rings given by polynomial optimization over a box. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 93-101. doi: 10.3934/naco.2014.4.93 |
[14] |
Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030 |
[15] |
M. S. Lee, B. S. Goh, H. G. Harno, K. H. Lim. On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 315-326. doi: 10.3934/naco.2018020 |
[16] |
Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411 |
[17] |
Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213 |
[18] |
Cheng-Dar Liou. Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method". Journal of Industrial & Management Optimization, 2012, 8 (3) : 727-732. doi: 10.3934/jimo.2012.8.727 |
[19] |
Kuo-Hsiung Wang, Chuen-Wen Liao, Tseng-Chang Yen. Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method. Journal of Industrial & Management Optimization, 2010, 6 (1) : 197-207. doi: 10.3934/jimo.2010.6.197 |
[20] |
Yong Xia. New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems. Journal of Industrial & Management Optimization, 2009, 5 (4) : 881-892. doi: 10.3934/jimo.2009.5.881 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]