# 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
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##### 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
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