April  2018, 14(2): 785-801. doi: 10.3934/jimo.2017075

A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization

Department of Applied Mathematics, Hainan University, Haikou 570228, China

* Corresponding author: Yigui Ou

Received  January 2017 Revised  March 2017 Published  September 2017

Fund Project: The work is supported by NNSF of China (No. 11261015) and NSF of Hainan Province (No. 2016CXTD004; No. 111001; No. 117011).

This paper presents a nonmonotone scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving nonsmooth convex optimization problems, which combines the idea of scaled memoryless BFGS preconditioned conjugate gradient method with the nonmonotone technique and the Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under mild conditions, the global convergence of the proposed method is established. Preliminary numerical results and related comparisons show that the proposed method can be applied to solve large scale nonsmooth convex optimization problems.

Citation: Yigui Ou, Xin Zhou. A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2018, 14 (2) : 785-801. doi: 10.3934/jimo.2017075
References:
[1]

N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Computational Optimization and Applications, 38 (2007), 401-416.  doi: 10.1007/s10589-007-9055-7.  Google Scholar

[2]

A. Auslender, Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study, 30 (1987), 102-126.   Google Scholar

[3]

S. Babaie-Kafaki, A modified scaled memoryless BFGS preconditioned conjugate gradient method for unconstrained optimization, 4OR, A Quarterly Journal of Operations Research, 11 (2013), 361-374.  doi: 10.1007/s10288-013-0233-4.  Google Scholar

[4]

S. Babaie-Kafaki and R. Chanbari, A class of descent four-term extension of the Dai-Liao conjugate gradient method based on the scaled memoryless BFGS update, Journal of Industrial and Management Optimization, 13 (2017), 649-658.  doi: 10.3934/jimo.2016038.  Google Scholar

[5]

J. Barzilai and J. M. Borwein, Two-point stepsize gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.   Google Scholar

[6]

E. Birgin and J. M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Applied Mathematics and Optimization, 43 (2001), 117-128.  doi: 10.1007/s00245-001-0003-0.  Google Scholar

[7]

J. F. BonnansJ. C. GilbertC. Lemarechal and C. Sagastizabal, A family of variable-metric proximal methods, Mathematical Programming, 68 (1995), 15-47.  doi: 10.1007/BF01585756.  Google Scholar

[8]

J. V. Burke and M. Qian, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Mathematical Programming, 88 (2000), 157-181.  doi: 10.1007/PL00011373.  Google Scholar

[9]

X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization, Mathematical Programming, 85 (1999), 313-334.  doi: 10.1007/s101070050059.  Google Scholar

[10]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, Serial A, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[11]

M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175.  doi: 10.1007/BF02591883.  Google Scholar

[12]

M. Fukushima and L. Q. Qi, A globally and superlinearly convergent algorithm for nonsmooth convex minimization, SIAM Journal on Optimization, 6 (1996), 1106-1120.  doi: 10.1137/S1052623494278839.  Google Scholar

[13]

M. HaaralaK. Miettinen and M. M. Mäkelä, New limited memory bundle method for large-scale nonsmooth optimization, Optimization Methods and Software, 19 (2004), 673-692.  doi: 10.1080/10556780410001689225.  Google Scholar

[14]

M. HaaralaK. Miettinen and M. M. Mäkelä, Globally convergent limited memory bundle method for large-scale nonsmooth optimization, Mathematical Programming, 109 (2007), 181-205.  doi: 10.1007/s10107-006-0728-2.  Google Scholar

[15]

W. W. Hager and H. C. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.   Google Scholar

[16]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.  Google Scholar

[17]

C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization, I: Theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385.   Google Scholar

[18]

Q. Li, Conjugate gradient type methods for the nondifferentiable convex minimization, Optimization Letters, 7 (2013), 533-545.   Google Scholar

[19]

D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optimization Methods and Software, 13 (2000), 181-201.   Google Scholar

[20]

S. LuZ. X. Wei and L. Li, A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization, Computational Optimization and Applications, 51 (2012), 551-573.   Google Scholar

[21]

L. Luk$\check{s}$an and J. Vl$\check{c}$ek, 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. Google Scholar

[22]

R. Mifflin, A quasi-second-order proximal bundle algorithm, Mathematical Programming, 73 (1996), 51-72.   Google Scholar

[23]

Y. G. Ou and H. C. Lin, An ODE-like nonmonotone method for nonsmooth convex optimization, Journal of Applied Mathematics and Computing, 52 (2016), 265-285.   Google Scholar

[24]

L. Q. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18 (1993), 227-244.  doi: 10.1287/moor.18.1.227.  Google Scholar

[25]

A. I. Rauf and M. Fukushima, A globally convergent BFGS method for nonsmooth convex optimization, Journal of Optimization Theory and Applications, 104 (2000), 539-558.   Google Scholar

[26]

N. Sagara and M. Fukushima, A trust region method for nonsmooth convex optimization, Journal of Industrial and Management Optimization, 1 (2005), 171-180.   Google Scholar

[27]

D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM Journal on Numerical Analysis, 15 (1978), 1247-1257.   Google Scholar

[28]

J. Shen, L. P. Pang and D. Li, An approximate quasi-Newton bundle-type method for nonsmooth optimization, Abstract and Applied Analysis, 2013, Art. ID 697474, 7 pp. doi: 10.1155/2013/697474.  Google Scholar

[29]

W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006.  Google Scholar

[30]

G. L. YuanZ. H. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large scale nonsmooth minimizations and nonlinear equations, Journal of Optimization Theory and Applications, 168 (2016), 129-152.   Google Scholar

[31]

G. L. Yuan, Z. Sheng and W. J. Liu, The modified HZ conjugate gradient algorithm for large scale nonsmooth optimization Plos One, 11(2016), e0164289, 15pp. doi: 10.1371/journal.pone.0164289.  Google Scholar

[32]

G. L. Yuan and Z. X. Wei, The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems, Mathematical Modelling and Analysis, 17 (2012), 203-216.   Google Scholar

[33]

G. L. YuanZ. X. Wei and G. Y. Li, A modified Polak-Ribiére-Polyak conjugate gradient algorithm for nonsmooth convex programs, Journal of Computational and Applied mathematics, 255 (2014), 86-96.   Google Scholar

[34]

G. L. YuanZ. X. Wei and Z. X. Wang, Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minization, Computational Optimization and Applications, 54 (2013), 45-64.   Google Scholar

[35]

H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM Jpournal on Optimization, 14 (2004), 1043-1056.  doi: 10.1137/S1052623403428208.  Google Scholar

[36]

L. ZhangW. J. Zhou and D. H. Li, A descent modified Polak-Ribiére-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640.  doi: 10.1093/imanum/drl016.  Google Scholar

show all references

References:
[1]

N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Computational Optimization and Applications, 38 (2007), 401-416.  doi: 10.1007/s10589-007-9055-7.  Google Scholar

[2]

A. Auslender, Numerical methods for nondifferentiable convex optimization, Mathematical Programming Study, 30 (1987), 102-126.   Google Scholar

[3]

S. Babaie-Kafaki, A modified scaled memoryless BFGS preconditioned conjugate gradient method for unconstrained optimization, 4OR, A Quarterly Journal of Operations Research, 11 (2013), 361-374.  doi: 10.1007/s10288-013-0233-4.  Google Scholar

[4]

S. Babaie-Kafaki and R. Chanbari, A class of descent four-term extension of the Dai-Liao conjugate gradient method based on the scaled memoryless BFGS update, Journal of Industrial and Management Optimization, 13 (2017), 649-658.  doi: 10.3934/jimo.2016038.  Google Scholar

[5]

J. Barzilai and J. M. Borwein, Two-point stepsize gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148.   Google Scholar

[6]

E. Birgin and J. M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Applied Mathematics and Optimization, 43 (2001), 117-128.  doi: 10.1007/s00245-001-0003-0.  Google Scholar

[7]

J. F. BonnansJ. C. GilbertC. Lemarechal and C. Sagastizabal, A family of variable-metric proximal methods, Mathematical Programming, 68 (1995), 15-47.  doi: 10.1007/BF01585756.  Google Scholar

[8]

J. V. Burke and M. Qian, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Mathematical Programming, 88 (2000), 157-181.  doi: 10.1007/PL00011373.  Google Scholar

[9]

X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization, Mathematical Programming, 85 (1999), 313-334.  doi: 10.1007/s101070050059.  Google Scholar

[10]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, Serial A, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[11]

M. Fukushima, A descent algorithm for nonsmooth convex optimization, Mathematical Programming, 30 (1984), 163-175.  doi: 10.1007/BF02591883.  Google Scholar

[12]

M. Fukushima and L. Q. Qi, A globally and superlinearly convergent algorithm for nonsmooth convex minimization, SIAM Journal on Optimization, 6 (1996), 1106-1120.  doi: 10.1137/S1052623494278839.  Google Scholar

[13]

M. HaaralaK. Miettinen and M. M. Mäkelä, New limited memory bundle method for large-scale nonsmooth optimization, Optimization Methods and Software, 19 (2004), 673-692.  doi: 10.1080/10556780410001689225.  Google Scholar

[14]

M. HaaralaK. Miettinen and M. M. Mäkelä, Globally convergent limited memory bundle method for large-scale nonsmooth optimization, Mathematical Programming, 109 (2007), 181-205.  doi: 10.1007/s10107-006-0728-2.  Google Scholar

[15]

W. W. Hager and H. C. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.   Google Scholar

[16]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.  Google Scholar

[17]

C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization, I: Theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385.   Google Scholar

[18]

Q. Li, Conjugate gradient type methods for the nondifferentiable convex minimization, Optimization Letters, 7 (2013), 533-545.   Google Scholar

[19]

D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optimization Methods and Software, 13 (2000), 181-201.   Google Scholar

[20]

S. LuZ. X. Wei and L. Li, A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization, Computational Optimization and Applications, 51 (2012), 551-573.   Google Scholar

[21]

L. Luk$\check{s}$an and J. Vl$\check{c}$ek, 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. Google Scholar

[22]

R. Mifflin, A quasi-second-order proximal bundle algorithm, Mathematical Programming, 73 (1996), 51-72.   Google Scholar

[23]

Y. G. Ou and H. C. Lin, An ODE-like nonmonotone method for nonsmooth convex optimization, Journal of Applied Mathematics and Computing, 52 (2016), 265-285.   Google Scholar

[24]

L. Q. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18 (1993), 227-244.  doi: 10.1287/moor.18.1.227.  Google Scholar

[25]

A. I. Rauf and M. Fukushima, A globally convergent BFGS method for nonsmooth convex optimization, Journal of Optimization Theory and Applications, 104 (2000), 539-558.   Google Scholar

[26]

N. Sagara and M. Fukushima, A trust region method for nonsmooth convex optimization, Journal of Industrial and Management Optimization, 1 (2005), 171-180.   Google Scholar

[27]

D. F. Shanno, On the convergence of a new conjugate gradient algorithm, SIAM Journal on Numerical Analysis, 15 (1978), 1247-1257.   Google Scholar

[28]

J. Shen, L. P. Pang and D. Li, An approximate quasi-Newton bundle-type method for nonsmooth optimization, Abstract and Applied Analysis, 2013, Art. ID 697474, 7 pp. doi: 10.1155/2013/697474.  Google Scholar

[29]

W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006.  Google Scholar

[30]

G. L. YuanZ. H. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large scale nonsmooth minimizations and nonlinear equations, Journal of Optimization Theory and Applications, 168 (2016), 129-152.   Google Scholar

[31]

G. L. Yuan, Z. Sheng and W. J. Liu, The modified HZ conjugate gradient algorithm for large scale nonsmooth optimization Plos One, 11(2016), e0164289, 15pp. doi: 10.1371/journal.pone.0164289.  Google Scholar

[32]

G. L. Yuan and Z. X. Wei, The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems, Mathematical Modelling and Analysis, 17 (2012), 203-216.   Google Scholar

[33]

G. L. YuanZ. X. Wei and G. Y. Li, A modified Polak-Ribiére-Polyak conjugate gradient algorithm for nonsmooth convex programs, Journal of Computational and Applied mathematics, 255 (2014), 86-96.   Google Scholar

[34]

G. L. YuanZ. X. Wei and Z. X. Wang, Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minization, Computational Optimization and Applications, 54 (2013), 45-64.   Google Scholar

[35]

H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM Jpournal on Optimization, 14 (2004), 1043-1056.  doi: 10.1137/S1052623403428208.  Google Scholar

[36]

L. ZhangW. J. Zhou and D. H. Li, A descent modified Polak-Ribiére-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26 (2006), 629-640.  doi: 10.1093/imanum/drl016.  Google Scholar

Figure 1.  Performance profiles based on iterations (left) and function evaluations (right)
Figure 2.  Performance profiles based on iterations (left) and function evaluations (right)
Table 1.  Testing functions for small scale problems
No.Functions$n$$x_0$${{f}_{ops}}\left( x \right)$
1Rosenbrock2(-1.2; 1)0
2Crescent2(-1.5; 2)0
3CB22(1; -0.1)1.9522245
4CB32(2; 2)2.0
5DEM2(1; 1)-3
6QL2(-1; 5)7.2
7LQ2(-0.5; -0.5)-1.4142136
8Mifflin 12(0.8; 0.6)-1.0
9Mifflin 22(-1; -1)-1.0
10Wolfe2(3; 2)-8
11Rosen-Suzuki4(0; 0; 0; 0)-44
12Shor5(0; 0; 0; 0; 1)22.600162
No.Functions$n$$x_0$${{f}_{ops}}\left( x \right)$
1Rosenbrock2(-1.2; 1)0
2Crescent2(-1.5; 2)0
3CB22(1; -0.1)1.9522245
4CB32(2; 2)2.0
5DEM2(1; 1)-3
6QL2(-1; 5)7.2
7LQ2(-0.5; -0.5)-1.4142136
8Mifflin 12(0.8; 0.6)-1.0
9Mifflin 22(-1; -1)-1.0
10Wolfe2(3; 2)-8
11Rosen-Suzuki4(0; 0; 0; 0)-44
12Shor5(0; 0; 0; 0; 1)22.600162
Table 2.  Numerical results for five algorithms
No.Algorithn 3.1LWTRYWBBSFTRRFBFGS
13/4/6/13/54/56/48/49/4/5/
2.6178e-90.2976e-73.4484e-77.1545e-46.2072e-10
23/4/3/7/14/16/31/32/35/36/
3.3026e-66.5430e-42.7450e-51.6000e-33.0590e-7
34/5/5/12/13/15/54/55/5/6/
1.95221.95221.95221.95731.9522
42/3/6/13/4/8/55/562/3/
2.00002.00002.00002.01002.0076
53/4/8/16/4/7/5/6/3/4
-3.0000-3.0000-3.0000-3.0000-3.0000
611/12/4/9/22/25/48/49/10/11/
7.20007.20007.20007.20037.2000
73/4/5/10/6/7/3/4/2/3/
-1.4142-1.4142-1.4142-1.4118-1.4033
834/35/15/31/3/6/59/60/57/58/
-1.0000-1.0000-0.9938-1.0000-1.0000
92/3/8/16/12/13/4/5/2/3/
-1.0000-1.0000-0.9999-0.9997-0.9813
103/4/12/24/9/12/43/46/4/5/
-8.0000-8.0000-8.0000-8.0000-8.0000
1149/106/20/40/8/9/60/61/25/31/
-43.9999-44.0000-43.9493-39.9924-43.9982
1214/16/14/28/9/10/71/72/66/152/
22.601922.600222.600422.689222.6017
No.Algorithn 3.1LWTRYWBBSFTRRFBFGS
13/4/6/13/54/56/48/49/4/5/
2.6178e-90.2976e-73.4484e-77.1545e-46.2072e-10
23/4/3/7/14/16/31/32/35/36/
3.3026e-66.5430e-42.7450e-51.6000e-33.0590e-7
34/5/5/12/13/15/54/55/5/6/
1.95221.95221.95221.95731.9522
42/3/6/13/4/8/55/562/3/
2.00002.00002.00002.01002.0076
53/4/8/16/4/7/5/6/3/4
-3.0000-3.0000-3.0000-3.0000-3.0000
611/12/4/9/22/25/48/49/10/11/
7.20007.20007.20007.20037.2000
73/4/5/10/6/7/3/4/2/3/
-1.4142-1.4142-1.4142-1.4118-1.4033
834/35/15/31/3/6/59/60/57/58/
-1.0000-1.0000-0.9938-1.0000-1.0000
92/3/8/16/12/13/4/5/2/3/
-1.0000-1.0000-0.9999-0.9997-0.9813
103/4/12/24/9/12/43/46/4/5/
-8.0000-8.0000-8.0000-8.0000-8.0000
1149/106/20/40/8/9/60/61/25/31/
-43.9999-44.0000-43.9493-39.9924-43.9982
1214/16/14/28/9/10/71/72/66/152/
22.601922.600222.600422.689222.6017
Table 3.  Testing functions for large scale problems
No.FunctionsInitial points $x_0$
1Generalization of MAXQ $(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2Generalization of MXHILB $(1, 1, \cdots, 1)$
3Chained LQ $(-0.5, -0.5, \cdots, -0.5)$
4Number of active faces $(1, 1, \cdots, 1)$
5Nonsmooth generalization of Brown 2 $(1, 0, 1, 0, \cdots)$
6Chained Mifflin 2 $(-1, -1, \cdots, -1)$
7Chained Crescent Ⅰ $(-1.5, 2, -1.5, 2, \cdots)$
8Chained Crescent Ⅱ $(1, 0, 1, 0 \cdots)$
No.FunctionsInitial points $x_0$
1Generalization of MAXQ $(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2Generalization of MXHILB $(1, 1, \cdots, 1)$
3Chained LQ $(-0.5, -0.5, \cdots, -0.5)$
4Number of active faces $(1, 1, \cdots, 1)$
5Nonsmooth generalization of Brown 2 $(1, 0, 1, 0, \cdots)$
6Chained Mifflin 2 $(-1, -1, \cdots, -1)$
7Chained Crescent Ⅰ $(-1.5, 2, -1.5, 2, \cdots)$
8Chained Crescent Ⅱ $(1, 0, 1, 0 \cdots)$
Table 4.  results for Algorithms 3.1 and CG-YWL
No.nAlgorithn 3.1CG-YWL
11000186/1601/2.6568e-10225/4710/6.9354e-8
5000242/2725/1.2183e-10250/5235/6.8798e-8
10000253/2997/3.4045e-10261/5466/6.6528e-8
2100094/1301/4.0582e-991/1482/8.2738e-9
5000119/1499/2.6731e-9111/1938/9.7206e-9
10000129/1901/2.1905e-9120/2127/5.8524e-9
3100037/110/2.3278e-937/114/7.2687e-9
500039/116/5.9987e-939/120/9.0932e-9
1000040/121/1.6943e-940/123/9.0941e-9
4100071/891/3.6735e-1177/1026/6.8037e-9
500082/937/7.9864e-1190/1281/7.8405e-9
1000086/1208/5.7163e-1196/1401/9.9366e-9
5100035/114/1.5021e-1138/117/7.2687e-9
500039/120/5.2352e-1140/123/9.0932e-9
1000041/126/2.1136e-1141/125/1.8188e-8
6100038/116/-5.3979e+337/114/-2.4975e+4
500041/123/-3.2153e+439/120/-1.2498e+5
1000043/129/-2.0127e+440/123/-2.4998e+5
7100034/101/7.0463e-1137/114/5.4897e-9
500036/113/3.8145e-1139/120/6.8294e-9
1000039/120/4.3654e-1140/123/6.8253e-9
8100037/112/6.0424e-1139/120/6.8185e-9
500039/121/1.8205e-1141/126/8.5258e-9
1000042/125/2.6473e-1142/129/8.5262e-9
No.nAlgorithn 3.1CG-YWL
11000186/1601/2.6568e-10225/4710/6.9354e-8
5000242/2725/1.2183e-10250/5235/6.8798e-8
10000253/2997/3.4045e-10261/5466/6.6528e-8
2100094/1301/4.0582e-991/1482/8.2738e-9
5000119/1499/2.6731e-9111/1938/9.7206e-9
10000129/1901/2.1905e-9120/2127/5.8524e-9
3100037/110/2.3278e-937/114/7.2687e-9
500039/116/5.9987e-939/120/9.0932e-9
1000040/121/1.6943e-940/123/9.0941e-9
4100071/891/3.6735e-1177/1026/6.8037e-9
500082/937/7.9864e-1190/1281/7.8405e-9
1000086/1208/5.7163e-1196/1401/9.9366e-9
5100035/114/1.5021e-1138/117/7.2687e-9
500039/120/5.2352e-1140/123/9.0932e-9
1000041/126/2.1136e-1141/125/1.8188e-8
6100038/116/-5.3979e+337/114/-2.4975e+4
500041/123/-3.2153e+439/120/-1.2498e+5
1000043/129/-2.0127e+440/123/-2.4998e+5
7100034/101/7.0463e-1137/114/5.4897e-9
500036/113/3.8145e-1139/120/6.8294e-9
1000039/120/4.3654e-1140/123/6.8253e-9
8100037/112/6.0424e-1139/120/6.8185e-9
500039/121/1.8205e-1141/126/8.5258e-9
1000042/125/2.6473e-1142/129/8.5262e-9
[1]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[2]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[3]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[4]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[5]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[6]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[7]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[8]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[9]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[10]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[11]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[12]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[13]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[14]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[15]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[16]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[17]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[18]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[19]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[20]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (155)
  • HTML views (814)
  • Cited by (0)

Other articles
by authors

[Back to Top]