# American Institute of Mathematical Sciences

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:

show all references

##### References:
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) 0.0000 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
 $n$ $x^*$ ${\tt f_{final}}$ ${\tt \|g^k\|_{final}}$ ${\tt Ni}$ ${\tt Time}$ 6 (-0.0000, 0.0000, -0.0000, 0.0000 0.0000, -0.0000) 0.0000 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
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) 0.0000 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
 ${\tt n}$ $x^*$ ${\tt f_{final}}$ ${\tt \|g^k\|_{final}}$ ${\tt Ni}$ ${\tt Time}$ 6 (0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000) 0.0000 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
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) 0.0000 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
 ${\tt n}$ $x^*$ ${\tt f_{final}}$ ${\tt \|g^k\|_{final}}$ ${\tt Ni}$ ${\tt Time}$ 6 (0.0000, -0.0000, -0.0000, 0.0000, -0.0000, 0.0000) 0.0000 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
 [1] Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069 [2] Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283 [3] Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171 [4] Hssaine Boualam, Ahmed Roubi. Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1897-1920. doi: 10.3934/jimo.2018128 [5] Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019135 [6] Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 [7] A. M. Bagirov, Moumita Ghosh, Dean Webb. A derivative-free method for linearly constrained nonsmooth optimization. Journal of Industrial & Management Optimization, 2006, 2 (3) : 319-338. doi: 10.3934/jimo.2006.2.319 [8] Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859 [9] Jueyou Li, Guoquan Li, Zhiyou Wu, Changzhi Wu, Xiangyu Wang, Jae-Myung Lee, Kwang-Hyo Jung. Incremental gradient-free method for nonsmooth distributed optimization. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1841-1857. doi: 10.3934/jimo.2017021 [10] Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789 [11] Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019100 [12] Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79 [13] 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 [14] Jian Gu, Xiantao Xiao, Liwei Zhang. A subgradient-based convex approximations method for DC programming and its applications. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1349-1366. doi: 10.3934/jimo.2016.12.1349 [15] Yan Gu, Nobuo Yamashita. A proximal ADMM with the Broyden family for convex optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020091 [16] Giancarlo Bigi. Componentwise versus global approaches to nonsmooth multiobjective optimization. Journal of Industrial & Management Optimization, 2005, 1 (1) : 21-32. doi: 10.3934/jimo.2005.1.21 [17] Fengqiu Liu, Xiaoping Xue. Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels. Journal of Industrial & Management Optimization, 2016, 12 (1) : 285-301. doi: 10.3934/jimo.2016.12.285 [18] David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253 [19] Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389 [20] Fan Jiang, Zhongming Wu, Xingju Cai. Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization. Journal of Industrial & Management Optimization, 2020, 16 (2) : 835-856. doi: 10.3934/jimo.2018181

2018 Impact Factor: 1.025