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Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach
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 |
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.
References:
[1] |
J. Baptiste, H. Urruty and C. Lemaéchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. |
[2] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. |
[3] |
R. Correa and C. Lemaréchal,
Convergence of some algorithms for convex minimization, Math. Program., 62 (1993), 261-275.
doi: 10.1007/BF01585170. |
[4] |
Z. Fu, K. Ren, J. Shu, X. Sun and F. Huang,
Enabling personalized search over encrypted out-sourced data with efficiency improvement, IEEE Transactions on Parallel and Distributed Systems, (2015).
|
[5] |
B. Gu, V. S. Sheng, K. Y. Tay, W. 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. |
[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. |
[7] |
J. Gu, X. 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. |
[8] |
K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lectures Notes in Mathematics, Springer, Berlin, 1985. |
[9] |
K. C. Kiwiel,
Proximity control in bundle methods for convex nondifferentiable minimization, Math. Program., 46 (1990), 105-122.
doi: 10.1007/BF01585731. |
[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. |
[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. |
[12] |
K. C. Kiwiel, T. 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. |
[13] |
K. C. Kiwiel,
Efficiency of proximal bundle methods, Journal of Optimization Theory and Applications, 104 (2000), 589-603.
doi: 10.1023/A:1004689609425. |
[14] |
J. Li, X. Li, B. Yang and X. Sun,
Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518.
|
[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. |
[16] |
J. J. Moreau, P. D. Panagiotopoulos and G. Strang (Eds. ), Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Basel, 1988. |
[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. |
[18] |
J. Outrata, M. KoÄvara and J. Zowe, Nonsmooth Approach to Optimization Problems With Equilibrium Constraints. Theory, Applications and Numerical Results, Kluwer Academic Publishers, Dordrecht, 1998. |
[19] |
Z. Pan, Y. Zhang and S. Kwong,
Efficient motion and disparity estimation optimization for low complexity multiview video coding, IEEE Transactions on Broadcasting, 61 (2015), 166-176.
|
[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. |
[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. |
[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. |
[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. |
[24] |
J. Shen, Z. 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. |
[25] |
K. Wang, L. Xu and D. Han,
A new parallel splitting descent method for structured variational inequalities, Journal of Industrial Management Optimization, 10 (2014), 461-476.
|
[26] |
Z. Xia, X. Wang, X. 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. |
[27] |
G. Yuan, Z. 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. |
[28] |
G. Yuan, Z. 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. |
[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. |
[30] |
J. Zhang, Y. 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. |
[31] |
J. Zhang, S. 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. |
show all references
References:
[1] |
J. Baptiste, H. Urruty and C. Lemaéchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, 1993. |
[2] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. |
[3] |
R. Correa and C. Lemaréchal,
Convergence of some algorithms for convex minimization, Math. Program., 62 (1993), 261-275.
doi: 10.1007/BF01585170. |
[4] |
Z. Fu, K. Ren, J. Shu, X. Sun and F. Huang,
Enabling personalized search over encrypted out-sourced data with efficiency improvement, IEEE Transactions on Parallel and Distributed Systems, (2015).
|
[5] |
B. Gu, V. S. Sheng, K. Y. Tay, W. 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. |
[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. |
[7] |
J. Gu, X. 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. |
[8] |
K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lectures Notes in Mathematics, Springer, Berlin, 1985. |
[9] |
K. C. Kiwiel,
Proximity control in bundle methods for convex nondifferentiable minimization, Math. Program., 46 (1990), 105-122.
doi: 10.1007/BF01585731. |
[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. |
[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. |
[12] |
K. C. Kiwiel, T. 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. |
[13] |
K. C. Kiwiel,
Efficiency of proximal bundle methods, Journal of Optimization Theory and Applications, 104 (2000), 589-603.
doi: 10.1023/A:1004689609425. |
[14] |
J. Li, X. Li, B. Yang and X. Sun,
Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518.
|
[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. |
[16] |
J. J. Moreau, P. D. Panagiotopoulos and G. Strang (Eds. ), Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Basel, 1988. |
[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. |
[18] |
J. Outrata, M. KoÄvara and J. Zowe, Nonsmooth Approach to Optimization Problems With Equilibrium Constraints. Theory, Applications and Numerical Results, Kluwer Academic Publishers, Dordrecht, 1998. |
[19] |
Z. Pan, Y. Zhang and S. Kwong,
Efficient motion and disparity estimation optimization for low complexity multiview video coding, IEEE Transactions on Broadcasting, 61 (2015), 166-176.
|
[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. |
[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. |
[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. |
[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. |
[24] |
J. Shen, Z. 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. |
[25] |
K. Wang, L. Xu and D. Han,
A new parallel splitting descent method for structured variational inequalities, Journal of Industrial Management Optimization, 10 (2014), 461-476.
|
[26] |
Z. Xia, X. Wang, X. 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. |
[27] |
G. Yuan, Z. 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. |
[28] |
G. Yuan, Z. 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. |
[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. |
[30] |
J. Zhang, Y. 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. |
[31] |
J. Zhang, S. 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. |
6 | (-0.0000, 0.0000, -0.0000, 0.0000 0.0000, -0.0000) | 0.0000 | 6.62e-07 | 13 | 1.1073 |
7 | 1.0e-05(0.12, -0.12, 0.42, 0.02, -0.16, -0.02, -0.06) | 2.01e-05 | 3.02e-06 | 16 | 1.596 |
8 | (0.0000, 0.0000, 0.0000, -0.0010, -0.0004, 0.0001, 0.0001, 0.0001) | 4.31e-7 | 3.40e-07 | 22 | 1.9153 |
9 | 1.0e-04(0.00, 0.032, 0.00, 0.00, -0.01, 0.01, 0.02, -0.01, -0.01) | 1.30e-05 | 4.10e-07 | 36 | 2.5103 |
10 | 1.0e-04(0.06, -0.07, -0.09, -0.16, 0.22, 0.25, 0.03, 0.29, -0.24, -0.20) | 3.26e-04 | 6.25e-07 | 36 | 1.8299 |
6 | (-0.0000, 0.0000, -0.0000, 0.0000 0.0000, -0.0000) | 0.0000 | 6.62e-07 | 13 | 1.1073 |
7 | 1.0e-05(0.12, -0.12, 0.42, 0.02, -0.16, -0.02, -0.06) | 2.01e-05 | 3.02e-06 | 16 | 1.596 |
8 | (0.0000, 0.0000, 0.0000, -0.0010, -0.0004, 0.0001, 0.0001, 0.0001) | 4.31e-7 | 3.40e-07 | 22 | 1.9153 |
9 | 1.0e-04(0.00, 0.032, 0.00, 0.00, -0.01, 0.01, 0.02, -0.01, -0.01) | 1.30e-05 | 4.10e-07 | 36 | 2.5103 |
10 | 1.0e-04(0.06, -0.07, -0.09, -0.16, 0.22, 0.25, 0.03, 0.29, -0.24, -0.20) | 3.26e-04 | 6.25e-07 | 36 | 1.8299 |
6 | (0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000) | 0.0000 | 1.06e-07 | 13 | 0.5323 |
7 | (-0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000) | 0.0000 | 1.03e-07 | 10 | 1.1167 |
8 | (-0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, 0.0000) | 0.0000 | 2.08e-08 | 27 | 2.1463 |
9 | 1.0e-05(0.01, 0.01, 0.01, 0.01, -0.02, -0.02, 0.01, 0.01, 0.02) | 1.37e-8 | 2.91e-07 | 36 | 2.6105 |
10 | 1.0e-06(-0.01, 0.02, -0.02, 0.02, -0.02, 0.01, -0.02, 0.02, 0.02, 0.01) | 4.32e-07 | 3.41e-07 | 39 | 3.3611 |
6 | (0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000) | 0.0000 | 1.06e-07 | 13 | 0.5323 |
7 | (-0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000) | 0.0000 | 1.03e-07 | 10 | 1.1167 |
8 | (-0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, -0.0000, 0.0000) | 0.0000 | 2.08e-08 | 27 | 2.1463 |
9 | 1.0e-05(0.01, 0.01, 0.01, 0.01, -0.02, -0.02, 0.01, 0.01, 0.02) | 1.37e-8 | 2.91e-07 | 36 | 2.6105 |
10 | 1.0e-06(-0.01, 0.02, -0.02, 0.02, -0.02, 0.01, -0.02, 0.02, 0.02, 0.01) | 4.32e-07 | 3.41e-07 | 39 | 3.3611 |
6 | (0.0000, -0.0000, -0.0000, 0.0000, -0.0000, 0.0000) | 0.0000 | 3.64e-07 | 16 | 1.0614 |
7 | (0.0000, 0.0000, -0.0000, 0.0000, -0.0000, -0.0000, 0.0000) | 0.0000 | 3.02e-07 | 19 | 1.9637 |
8 | (0.0000, -0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000) | 0.0000 | 4.01e-07 | 26 | 1.9437 |
9 | 1.0e-06(0.03, 0.03, 0.01, -0.01, 0.00, 0.00, 0.01, -0.01, -0.01) | 2.07e-7 | 2.14e-07 | 33 | 2.8025 |
10 | 1.0e-06(0.02, -0.02, -0.01, 0.04, -0.02, 0.04, 0.02, -0.02, -0.02, 0.01) | 4.30e-08 | 7.19e-08 | 41 | 3.3061 |
6 | (0.0000, -0.0000, -0.0000, 0.0000, -0.0000, 0.0000) | 0.0000 | 3.64e-07 | 16 | 1.0614 |
7 | (0.0000, 0.0000, -0.0000, 0.0000, -0.0000, -0.0000, 0.0000) | 0.0000 | 3.02e-07 | 19 | 1.9637 |
8 | (0.0000, -0.0000, 0.0000, 0.0000, -0.0000, 0.0000, 0.0000, 0.0000) | 0.0000 | 4.01e-07 | 26 | 1.9437 |
9 | 1.0e-06(0.03, 0.03, 0.01, -0.01, 0.00, 0.00, 0.01, -0.01, -0.01) | 2.07e-7 | 2.14e-07 | 33 | 2.8025 |
10 | 1.0e-06(0.02, -0.02, -0.01, 0.04, -0.02, 0.04, 0.02, -0.02, -0.02, 0.01) | 4.30e-08 | 7.19e-08 | 41 | 3.3061 |
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