American Institute of Mathematical Sciences

Advanced
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

Jie Shen 1,, , Jian Lv 2,3, , Fang-Fang Guo 2,3, , Ya-Li Gao 1, and  Rui Zhao 1,

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.

Keywords: Nonsmooth optimization, proximal bundle method, subgradient, localization set, Chebychev center.
Mathematics Subject Classification: Primary: 90C25; Secondary: 65K05.
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. 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).   Google Scholar

[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.  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. 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.  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. 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.  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. 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.   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. 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.   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. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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).   Google Scholar

[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.  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. 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.  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. 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.  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. 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.   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. 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.   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. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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
Table Options

Download as excel

$ 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
Table Options

Download as excel

${\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
Table Options

Download as excel

${\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
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