July  2018, 14(3): 1143-1155. doi: 10.3934/jimo.2018003

A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence

1. 

School of Mathematics, Liaoning Normal University, Dalian, 116029, China

2. 

School of Finance, Zhejiang University of Finance and Economics, Hangzhou, 310018, China

3. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

* Corresponding author: Jie Shen

Received  April 2016 Revised  August 2017 Published  January 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China under Project No. 11301246, No. 11671183, No. 11601061 and the Natural Science Foundation Plan Project of Liaoning Province No.20170540573, the Foundation of Educational Committee of Liaoning Province No.LF201783607 and the Fundamental Research Funds for the Central Universities of China No.DUT16LK07.

Motivated by the proximal-like bundle method [K. C. Kiwiel, Journal of Optimization Theory and Applications, 104(3) (2000), 589-603.], we establish a new proximal Chebychev center cutting plane algorithm for a type of nonsmooth optimization problems. At each step of the algorithm, a new optimality measure is investigated instead of the classical optimality measure. The convergence analysis shows that an $\varepsilon$-optimal solution can be obtained within $O(1/\varepsilon^3)$ iterations. The numerical result is presented to show the validity of the conclusion and it shows that the method is competitive to the classical proximal-like bundle method.

Citation: Jie Shen, Jian Lv, Fang-Fang Guo, Ya-Li Gao, Rui Zhao. A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1143-1155. doi: 10.3934/jimo.2018003
References:
[1]

J. Baptiste, H. Urruty and C. Lemaéchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. Google Scholar

[2]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[3]

R. Correa and C. Lemaréchal, Convergence of some algorithms for convex minimization, Math. Program., 62 (1993), 261-275.  doi: 10.1007/BF01585170.  Google Scholar

[4]

Z. FuK. RenJ. ShuX. Sun and F. Huang, Enabling personalized search over encrypted out-sourced data with efficiency improvement, IEEE Transactions on Parallel and Distributed Systems, (2015).   Google Scholar

[5]

B. GuV. S. ShengK. Y. TayW. Romano and S. Li, Incremental support vector learning for ordinal regression, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 1403-1416.  doi: 10.1109/TNNLS.2014.2342533.  Google Scholar

[6]

B. Gu and V. S. Sheng, A robust regularization path algorithm for ν-support vector classification, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 1241-1248.  doi: 10.1109/TNNLS.2016.2527796.  Google Scholar

[7]

J. GuX. Xiao and L. Zhang, A subgradient-based convex approximations method for DC programming and its applications, Journal of Industrial Management Optimization, 12 (2016), 1349-1366.  doi: 10.3934/jimo.2016.12.1349.  Google Scholar

[8]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lectures Notes in Mathematics, Springer, Berlin, 1985.  Google Scholar

[9]

K. C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Math. Program., 46 (1990), 105-122.  doi: 10.1007/BF01585731.  Google Scholar

[10]

K. C. Kiwiel, Efficiency of the analytic center cutting plane method for convex minimization, SIAM J. Optim., 7 (1997), 336-346.  doi: 10.1137/S1052623494275768.  Google Scholar

[11]

K. C. Kiwiel, The efficiency of subgradient projection methods for convex optimization. Part 1: General level methods, SIAM Journal on Control and Optimization, 34 (1996), 660-676.  doi: 10.1137/0334031.  Google Scholar

[12]

K. C. KiwielT. Larsson and P. O. Lindberg, The efficiency of ball step subgradient level methods for convex optimization, Mathematics of Operations Research, 24 (1999), 237-254.  doi: 10.1287/moor.24.1.237.  Google Scholar

[13]

K. C. Kiwiel, Efficiency of proximal bundle methods, Journal of Optimization Theory and Applications, 104 (2000), 589-603.  doi: 10.1023/A:1004689609425.  Google Scholar

[14]

J. LiX. LiB. Yang and X. Sun, Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518.   Google Scholar

[15]

E. S. Mistakidis and G. E. Stavroulakis, Nonconvex Optimization in Mechanics. Smooth and Nonsmooth Algorithms, Heuristics and Engineering Applications, F. E. M. Kluwer Academic Publisher, Dordrecht, 1998.  Google Scholar

[16]

J. J. Moreau, P. D. Panagiotopoulos and G. Strang (Eds. ), Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Basel, 1988.  Google Scholar

[17]

A. Ouorou, A proximal cutting plane method using Chebychev center for nonsmooth convex optimization, Math. Program. Ser. A, 119 (2009), 239-271.  doi: 10.1007/s10107-008-0209-x.  Google Scholar

[18]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems With Equilibrium Constraints. Theory, Applications and Numerical Results, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar

[19]

Z. PanY. Zhang and S. Kwong, Efficient motion and disparity estimation optimization for low complexity multiview video coding, IEEE Transactions on Broadcasting, 61 (2015), 166-176.   Google Scholar

[20]

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 (1992), 121-152.  doi: 10.1137/0802008.  Google Scholar

[21]

J. Shen, D. Li and L. Pang, A cutting plane and level stabilization bundle method with inexact data for minimizing nonsmooth nonconvex functions, Abstract and Applied Analysis, 2014 (2014), Article ID 192893, 6pp.  Google Scholar

[22]

J. Shen and L. Pang, An approximate bundle-type auxiliary problem method for generalized variational inequality, Mathematical and Computer Modeling, 48 (2008), 769-775.  doi: 10.1016/j.mcm.2007.11.005.  Google Scholar

[23]

J. Shen, X. Liu, F. Guo and S. Wang, An approximate redistributed proximal bundle method with inexact data for minimizing nonsmooth nonconvex functions, Mathematical Problems in Engineering, 2015 (2015), Article ID 215310, 9pp.  Google Scholar

[24]

J. ShenZ. Xia and L. Pang, A proximal bundle method with inexact data for convex nondifferentiable minimization, Nonlinear Analysis A : theory, method and applications, 66 (2007), 2016-2027.  doi: 10.1016/j.na.2006.02.039.  Google Scholar

[25]

K. WangL. Xu and D. Han, A new parallel splitting descent method for structured variational inequalities, Journal of Industrial Management Optimization, 10 (2014), 461-476.   Google Scholar

[26]

Z. XiaX. WangX. Sun and Q. Wang, A secure and dynamic multi-keyword ranked search scheme over encrypted cloud data, IEEE Transactions on Parallel and Distributed Systems, 27 (2016), 340-352.  doi: 10.1109/TPDS.2015.2401003.  Google Scholar

[27]

G. YuanZ. Wei and G. Li, A modified Polak-Ribiére-Polyak conjugate gradient algorithm for nonsmooth convex programs, Journal of Computational and Applied Mathematics, 255 (2014), 86-96.  doi: 10.1016/j.cam.2013.04.032.  Google Scholar

[28]

G. YuanZ. 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.  doi: 10.1007/s10957-015-0781-1.  Google Scholar

[29]

G. Yuan and M. Zhang, A three-terms Polak-Ribiére-Polyak conjugate gradient algorithm for large-scale nonlinear equations, Journal of Computational and Applied Mathematics, 286 (2015), 186-195.  doi: 10.1016/j.cam.2015.03.014.  Google Scholar

[30]

J. ZhangY. Li and L. Zhang, On the coderivative of the solution mapping to a second-order cone constrained parametric variational inequality, Journal of Global Optimization, 61 (2015), 379-396.  doi: 10.1007/s10898-014-0181-3.  Google Scholar

[31]

J. ZhangS. Lin and L. Zhang, A log-exponential regularization method for a mathmatical program with general vertical complementarity constraints, Journal of Industrial Management Optimization, 9 (2013), 561-577.  doi: 10.3934/jimo.2013.9.561.  Google Scholar

show all references

References:
[1]

J. Baptiste, H. Urruty and C. Lemaéchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. Google Scholar

[2]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[3]

R. Correa and C. Lemaréchal, Convergence of some algorithms for convex minimization, Math. Program., 62 (1993), 261-275.  doi: 10.1007/BF01585170.  Google Scholar

[4]

Z. FuK. RenJ. ShuX. Sun and F. Huang, Enabling personalized search over encrypted out-sourced data with efficiency improvement, IEEE Transactions on Parallel and Distributed Systems, (2015).   Google Scholar

[5]

B. GuV. S. ShengK. Y. TayW. Romano and S. Li, Incremental support vector learning for ordinal regression, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 1403-1416.  doi: 10.1109/TNNLS.2014.2342533.  Google Scholar

[6]

B. Gu and V. S. Sheng, A robust regularization path algorithm for ν-support vector classification, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 1241-1248.  doi: 10.1109/TNNLS.2016.2527796.  Google Scholar

[7]

J. GuX. Xiao and L. Zhang, A subgradient-based convex approximations method for DC programming and its applications, Journal of Industrial Management Optimization, 12 (2016), 1349-1366.  doi: 10.3934/jimo.2016.12.1349.  Google Scholar

[8]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lectures Notes in Mathematics, Springer, Berlin, 1985.  Google Scholar

[9]

K. C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Math. Program., 46 (1990), 105-122.  doi: 10.1007/BF01585731.  Google Scholar

[10]

K. C. Kiwiel, Efficiency of the analytic center cutting plane method for convex minimization, SIAM J. Optim., 7 (1997), 336-346.  doi: 10.1137/S1052623494275768.  Google Scholar

[11]

K. C. Kiwiel, The efficiency of subgradient projection methods for convex optimization. Part 1: General level methods, SIAM Journal on Control and Optimization, 34 (1996), 660-676.  doi: 10.1137/0334031.  Google Scholar

[12]

K. C. KiwielT. Larsson and P. O. Lindberg, The efficiency of ball step subgradient level methods for convex optimization, Mathematics of Operations Research, 24 (1999), 237-254.  doi: 10.1287/moor.24.1.237.  Google Scholar

[13]

K. C. Kiwiel, Efficiency of proximal bundle methods, Journal of Optimization Theory and Applications, 104 (2000), 589-603.  doi: 10.1023/A:1004689609425.  Google Scholar

[14]

J. LiX. LiB. Yang and X. Sun, Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518.   Google Scholar

[15]

E. S. Mistakidis and G. E. Stavroulakis, Nonconvex Optimization in Mechanics. Smooth and Nonsmooth Algorithms, Heuristics and Engineering Applications, F. E. M. Kluwer Academic Publisher, Dordrecht, 1998.  Google Scholar

[16]

J. J. Moreau, P. D. Panagiotopoulos and G. Strang (Eds. ), Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Basel, 1988.  Google Scholar

[17]

A. Ouorou, A proximal cutting plane method using Chebychev center for nonsmooth convex optimization, Math. Program. Ser. A, 119 (2009), 239-271.  doi: 10.1007/s10107-008-0209-x.  Google Scholar

[18]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems With Equilibrium Constraints. Theory, Applications and Numerical Results, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar

[19]

Z. PanY. Zhang and S. Kwong, Efficient motion and disparity estimation optimization for low complexity multiview video coding, IEEE Transactions on Broadcasting, 61 (2015), 166-176.   Google Scholar

[20]

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 (1992), 121-152.  doi: 10.1137/0802008.  Google Scholar

[21]

J. Shen, D. Li and L. Pang, A cutting plane and level stabilization bundle method with inexact data for minimizing nonsmooth nonconvex functions, Abstract and Applied Analysis, 2014 (2014), Article ID 192893, 6pp.  Google Scholar

[22]

J. Shen and L. Pang, An approximate bundle-type auxiliary problem method for generalized variational inequality, Mathematical and Computer Modeling, 48 (2008), 769-775.  doi: 10.1016/j.mcm.2007.11.005.  Google Scholar

[23]

J. Shen, X. Liu, F. Guo and S. Wang, An approximate redistributed proximal bundle method with inexact data for minimizing nonsmooth nonconvex functions, Mathematical Problems in Engineering, 2015 (2015), Article ID 215310, 9pp.  Google Scholar

[24]

J. ShenZ. Xia and L. Pang, A proximal bundle method with inexact data for convex nondifferentiable minimization, Nonlinear Analysis A : theory, method and applications, 66 (2007), 2016-2027.  doi: 10.1016/j.na.2006.02.039.  Google Scholar

[25]

K. WangL. Xu and D. Han, A new parallel splitting descent method for structured variational inequalities, Journal of Industrial Management Optimization, 10 (2014), 461-476.   Google Scholar

[26]

Z. XiaX. WangX. Sun and Q. Wang, A secure and dynamic multi-keyword ranked search scheme over encrypted cloud data, IEEE Transactions on Parallel and Distributed Systems, 27 (2016), 340-352.  doi: 10.1109/TPDS.2015.2401003.  Google Scholar

[27]

G. YuanZ. Wei and G. Li, A modified Polak-Ribiére-Polyak conjugate gradient algorithm for nonsmooth convex programs, Journal of Computational and Applied Mathematics, 255 (2014), 86-96.  doi: 10.1016/j.cam.2013.04.032.  Google Scholar

[28]

G. YuanZ. 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.  doi: 10.1007/s10957-015-0781-1.  Google Scholar

[29]

G. Yuan and M. Zhang, A three-terms Polak-Ribiére-Polyak conjugate gradient algorithm for large-scale nonlinear equations, Journal of Computational and Applied Mathematics, 286 (2015), 186-195.  doi: 10.1016/j.cam.2015.03.014.  Google Scholar

[30]

J. ZhangY. Li and L. Zhang, On the coderivative of the solution mapping to a second-order cone constrained parametric variational inequality, Journal of Global Optimization, 61 (2015), 379-396.  doi: 10.1007/s10898-014-0181-3.  Google Scholar

[31]

J. ZhangS. Lin and L. Zhang, A log-exponential regularization method for a mathmatical program with general vertical complementarity constraints, Journal of Industrial Management Optimization, 9 (2013), 561-577.  doi: 10.3934/jimo.2013.9.561.  Google Scholar

Table 1.  Test results obtained by $pc^3pa$ algorithm for $\min\limits_{x \in R^n} f_{1}(x)$
$ n$$x^*$${\tt f_{final}}$${\tt \|g^k\|_{final}}$$ {\tt Ni}$$ {\tt Time}$
6(-0.0000, 0.0000, -0.0000, 0.0000 0.0000, -0.0000)0.00006.62e-07131.1073
71.0e-05(0.12, -0.12, 0.42, 0.02, -0.16, -0.02, -0.06)2.01e-053.02e-06161.596
8(0.0000, 0.0000, 0.0000, -0.0010, -0.0004, 0.0001, 0.0001, 0.0001)4.31e-73.40e-07221.9153
91.0e-04(0.00, 0.032, 0.00, 0.00, -0.01, 0.01, 0.02, -0.01, -0.01)1.30e-054.10e-07362.5103
101.0e-04(0.06, -0.07, -0.09, -0.16, 0.22, 0.25, 0.03, 0.29, -0.24, -0.20)3.26e-046.25e-07361.8299
$ n$$x^*$${\tt f_{final}}$${\tt \|g^k\|_{final}}$$ {\tt Ni}$$ {\tt Time}$
6(-0.0000, 0.0000, -0.0000, 0.0000 0.0000, -0.0000)0.00006.62e-07131.1073
71.0e-05(0.12, -0.12, 0.42, 0.02, -0.16, -0.02, -0.06)2.01e-053.02e-06161.596
8(0.0000, 0.0000, 0.0000, -0.0010, -0.0004, 0.0001, 0.0001, 0.0001)4.31e-73.40e-07221.9153
91.0e-04(0.00, 0.032, 0.00, 0.00, -0.01, 0.01, 0.02, -0.01, -0.01)1.30e-054.10e-07362.5103
101.0e-04(0.06, -0.07, -0.09, -0.16, 0.22, 0.25, 0.03, 0.29, -0.24, -0.20)3.26e-046.25e-07361.8299
Table 2.  Test results obtained by $pc^3pa$ algorithm for $\min\limits_{x \in R^n} f_{2}(x)$
${\tt n} $$x^*$${\tt f_{final}}$${\tt \|g^k\|_{final}}$$ {\tt Ni}$${\tt Time} $
6(0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000)0.00001.06e-07130.5323
7(-0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000)0.00001.03e-07101.1167
8(-0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, 0.0000)0.00002.08e-08272.1463
91.0e-05(0.01, 0.01, 0.01, 0.01, -0.02, -0.02, 0.01, 0.01, 0.02)1.37e-82.91e-07362.6105
101.0e-06(-0.01, 0.02, -0.02, 0.02, -0.02, 0.01, -0.02, 0.02, 0.02, 0.01)4.32e-073.41e-07393.3611
${\tt n} $$x^*$${\tt f_{final}}$${\tt \|g^k\|_{final}}$$ {\tt Ni}$${\tt Time} $
6(0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000)0.00001.06e-07130.5323
7(-0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000)0.00001.03e-07101.1167
8(-0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, 0.0000)0.00002.08e-08272.1463
91.0e-05(0.01, 0.01, 0.01, 0.01, -0.02, -0.02, 0.01, 0.01, 0.02)1.37e-82.91e-07362.6105
101.0e-06(-0.01, 0.02, -0.02, 0.02, -0.02, 0.01, -0.02, 0.02, 0.02, 0.01)4.32e-073.41e-07393.3611
Table 3.  Test results obtained by $pc^3pa$ algorithm for $\min\limits_{x \in R^n} f_{3}(x)$
${\tt n} $$x^*$${\tt f_{final}}$${\tt \|g^k\|_{final}}$${\tt Ni} $${\tt Time} $
6(0.0000, -0.0000, -0.0000, 0.0000, -0.0000, 0.0000)0.00003.64e-07161.0614
7(0.0000, 0.0000, -0.0000, 0.0000, -0.0000, -0.0000, 0.0000)0.00003.02e-07191.9637
8(0.0000, -0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000)0.00004.01e-07261.9437
91.0e-06(0.03, 0.03, 0.01, -0.01, 0.00, 0.00, 0.01, -0.01, -0.01)2.07e-72.14e-07332.8025
101.0e-06(0.02, -0.02, -0.01, 0.04, -0.02, 0.04, 0.02, -0.02, -0.02, 0.01)4.30e-087.19e-08413.3061
${\tt n} $$x^*$${\tt f_{final}}$${\tt \|g^k\|_{final}}$${\tt Ni} $${\tt Time} $
6(0.0000, -0.0000, -0.0000, 0.0000, -0.0000, 0.0000)0.00003.64e-07161.0614
7(0.0000, 0.0000, -0.0000, 0.0000, -0.0000, -0.0000, 0.0000)0.00003.02e-07191.9637
8(0.0000, -0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000)0.00004.01e-07261.9437
91.0e-06(0.03, 0.03, 0.01, -0.01, 0.00, 0.00, 0.01, -0.01, -0.01)2.07e-72.14e-07332.8025
101.0e-06(0.02, -0.02, -0.01, 0.04, -0.02, 0.04, 0.02, -0.02, -0.02, 0.01)4.30e-087.19e-08413.3061
[1]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

[2]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[3]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[4]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[5]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[6]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[7]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[8]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[9]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[10]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[11]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

[12]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

[13]

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

[14]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[15]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[16]

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

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (149)
  • HTML views (911)
  • Cited by (0)

Other articles
by authors

[Back to Top]