# American Institute of Mathematical Sciences

2016, 6(2): 161-173. doi: 10.3934/naco.2016006

## A new smoothing approach to exact penalty functions for inequality constrained optimization problems

 1 Suleyman Demirel University, Department of Mathematics, Isparta, 32100, Turkey, Turkey, Turkey

Received  December 2015 Revised  May 2016 Published  June 2016

In this study, we introduce a new smoothing approximation to the non-differentiable exact penalty functions for inequality constrained optimization problems. Error estimations are investigated between non-smooth penalty function and smoothed penalty function. In order to demonstrate the effectiveness of proposed smoothing approach the numerical examples are given.
Citation: Ahmet Sahiner, Gulden Kapusuz, Nurullah Yilmaz. A new smoothing approach to exact penalty functions for inequality constrained optimization problems. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 161-173. doi: 10.3934/naco.2016006
##### References:
 [1] A. M. Bagirov, A. Al Nuamiat and N. Sultanova, Hyperbolic smoothing functions for nonsmooth minimization,, Optimization, 62 (2013), 759. doi: 10.1080/02331934.2012.675335. Google Scholar [2] F. S. Bai, Z. Y. Wu and D. L. Zhu, Lower order calmness and exact penalty fucntion,, Optimization Methods ans Software, 21 (2006), 515. doi: 10.1080/10556780600627693. Google Scholar [3] A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems,, Lecture notes in mathematics, (1989), 1. Google Scholar [4] D. Bertsekas, Nondifferentiable optimization via approximation,, Mathematical Programming Study, 3 (1975), 1. Google Scholar [5] C. Chen and O. L. Mangasarian, A Class of smoothing functions for nonlinear and mixed complementarity problem,, Computational Optimization and Application, 5 (1996), 97. doi: 10.1007/BF00249052. Google Scholar [6] X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation,, Mathematical Programming Serie B, 134 (2012), 71. doi: 10.1007/s10107-012-0569-0. Google Scholar [7] S. J. Lian, Smoothing approximation to l1 exact penalty for inequality constrained optimization,, Applied Mathematics and Computation, 219 (2012), 3113. doi: 10.1016/j.amc.2012.09.042. Google Scholar [8] B. Liu, On smoothing exact penalty function for nonlinear constrained optimization problem,, Journal of Applied Mathematics and Computing, 30 (2009), 259. doi: 10.1007/s12190-008-0171-z. Google Scholar [9] M. C. Pinar and S. Zenios, On smoothing exact penalty functions for convex constrained optimization,, SIAM Journal on Optimization, 4 (1994), 468. doi: 10.1137/0804027. Google Scholar [10] Z. Meng, C. Dang, M. Jiang and R. Shen, A smoothing objective penalty function algorithm foe inequality constrained optimization problems,, Numerical Functional Analysis and Optimization, 32 (2011), 806. doi: 10.1080/01630563.2011.577262. Google Scholar [11] Z. Y. Wu, H. W. J. Lee, F. S. Bai and L. S. Zhang, Quadratic smoothing approximation to l1 exact penalty function in global optimization,, Journal of Industrail and Management Optimization, 53 (2005), 533. doi: 10.3934/jimo.2005.1.533. Google Scholar [12] Z. Y. Wu, F. S. Bai, X. Q. Yang and L. S. Zhang, An exact lower orderpenalty function and its smoothing in nonlinear programming,, Optimization, 53 (2004), 51. doi: 10.1080/02331930410001662199. Google Scholar [13] A. E. Xavier, The hyperbolic smoothing clustering method,, Pattern Recognition, 43 (2010), 731. Google Scholar [14] A. E. Xavier,A. A. F. D. Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing,, Journal of Global Optimization, 31 (2005), 493. doi: 10.1007/s10898-004-0737-8. Google Scholar [15] X. Xu, Z. Meng, J. Sun and R. Shen, A penalty function method based on smoothing lower order penalty function,, Journal of Computational and Applied Mathematics, 235 (2011), 4047. doi: 10.1016/j.cam.2011.02.031. Google Scholar [16] N. Yilmaz, A. Sahiner, A New Global Optimization Technique Based on the Smoothing Approach for Non-smooth, Non-convex Optimization,, Submitted., (). Google Scholar [17] N. Yilmaz, A. Sahiner, Smoothing Approach for Non-lipschitz Optimization,, Submitted., (). Google Scholar [18] I. Zang, A smooting out technique for min-max optimization,, Mathematical Programming, 19 (1980), 61. doi: 10.1007/BF01581628. Google Scholar [19] W. I. Zangwill, Nonlinear programing via penalty functions,, Management Science, 13 (1967), 344. Google Scholar

show all references

##### References:
 [1] A. M. Bagirov, A. Al Nuamiat and N. Sultanova, Hyperbolic smoothing functions for nonsmooth minimization,, Optimization, 62 (2013), 759. doi: 10.1080/02331934.2012.675335. Google Scholar [2] F. S. Bai, Z. Y. Wu and D. L. Zhu, Lower order calmness and exact penalty fucntion,, Optimization Methods ans Software, 21 (2006), 515. doi: 10.1080/10556780600627693. Google Scholar [3] A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems,, Lecture notes in mathematics, (1989), 1. Google Scholar [4] D. Bertsekas, Nondifferentiable optimization via approximation,, Mathematical Programming Study, 3 (1975), 1. Google Scholar [5] C. Chen and O. L. Mangasarian, A Class of smoothing functions for nonlinear and mixed complementarity problem,, Computational Optimization and Application, 5 (1996), 97. doi: 10.1007/BF00249052. Google Scholar [6] X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation,, Mathematical Programming Serie B, 134 (2012), 71. doi: 10.1007/s10107-012-0569-0. Google Scholar [7] S. J. Lian, Smoothing approximation to l1 exact penalty for inequality constrained optimization,, Applied Mathematics and Computation, 219 (2012), 3113. doi: 10.1016/j.amc.2012.09.042. Google Scholar [8] B. Liu, On smoothing exact penalty function for nonlinear constrained optimization problem,, Journal of Applied Mathematics and Computing, 30 (2009), 259. doi: 10.1007/s12190-008-0171-z. Google Scholar [9] M. C. Pinar and S. Zenios, On smoothing exact penalty functions for convex constrained optimization,, SIAM Journal on Optimization, 4 (1994), 468. doi: 10.1137/0804027. Google Scholar [10] Z. Meng, C. Dang, M. Jiang and R. Shen, A smoothing objective penalty function algorithm foe inequality constrained optimization problems,, Numerical Functional Analysis and Optimization, 32 (2011), 806. doi: 10.1080/01630563.2011.577262. Google Scholar [11] Z. Y. Wu, H. W. J. Lee, F. S. Bai and L. S. Zhang, Quadratic smoothing approximation to l1 exact penalty function in global optimization,, Journal of Industrail and Management Optimization, 53 (2005), 533. doi: 10.3934/jimo.2005.1.533. Google Scholar [12] Z. Y. Wu, F. S. Bai, X. Q. Yang and L. S. Zhang, An exact lower orderpenalty function and its smoothing in nonlinear programming,, Optimization, 53 (2004), 51. doi: 10.1080/02331930410001662199. Google Scholar [13] A. E. Xavier, The hyperbolic smoothing clustering method,, Pattern Recognition, 43 (2010), 731. Google Scholar [14] A. E. Xavier,A. A. F. D. Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing,, Journal of Global Optimization, 31 (2005), 493. doi: 10.1007/s10898-004-0737-8. Google Scholar [15] X. Xu, Z. Meng, J. Sun and R. Shen, A penalty function method based on smoothing lower order penalty function,, Journal of Computational and Applied Mathematics, 235 (2011), 4047. doi: 10.1016/j.cam.2011.02.031. Google Scholar [16] N. Yilmaz, A. Sahiner, A New Global Optimization Technique Based on the Smoothing Approach for Non-smooth, Non-convex Optimization,, Submitted., (). Google Scholar [17] N. Yilmaz, A. Sahiner, Smoothing Approach for Non-lipschitz Optimization,, Submitted., (). Google Scholar [18] I. Zang, A smooting out technique for min-max optimization,, Mathematical Programming, 19 (1980), 61. doi: 10.1007/BF01581628. Google Scholar [19] W. I. Zangwill, Nonlinear programing via penalty functions,, Management Science, 13 (1967), 344. Google Scholar
 [1] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895 [2] Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial & Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533 [3] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485 [4] Zhongwen Chen, Songqiang Qiu, Yujie Jiao. A penalty-free method for equality constrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (2) : 391-409. doi: 10.3934/jimo.2013.9.391 [5] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [6] Na Zhao, Zheng-Hai Huang. A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function. Journal of Industrial & Management Optimization, 2011, 7 (2) : 467-482. doi: 10.3934/jimo.2011.7.467 [7] Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949 [8] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [9] Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012 [10] Xiaodi Bai, Xiaojin Zheng, Xiaoling Sun. A survey on probabilistically constrained optimization problems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 767-778. doi: 10.3934/naco.2012.2.767 [11] Ahmet Sahiner, Nurullah Yilmaz, Gulden Kapusuz. A novel modeling and smoothing technique in global optimization. Journal of Industrial & Management Optimization, 2019, 15 (1) : 113-130. doi: 10.3934/jimo.2018035 [12] Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 [13] Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial & Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585 [14] Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353 [15] Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 529-537. doi: 10.3934/naco.2011.1.529 [16] Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006 [17] Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial & Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435 [18] Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial & Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783 [19] Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 965-976. doi: 10.3934/jimo.2014.10.965 [20] X. X. Huang, Xiaoqi Yang, K. L. Teo. A smoothing scheme for optimization problems with Max-Min constraints. Journal of Industrial & Management Optimization, 2007, 3 (2) : 209-222. doi: 10.3934/jimo.2007.3.209

Impact Factor: