American Institute of Mathematical Sciences

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

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

Ahmet Sahiner 1, , Gulden Kapusuz 1, and  Nurullah Yilmaz 1,

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.
Keywords: Constrained optimization, penalty function, smoothing approach..
Mathematics Subject Classification: Primary: 90C30, 57R12; Secondary: 53C3.
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), 759-782. 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), 515-525. doi: 10.1080/10556780600627693.  Google Scholar

[3]

A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems, Lecture notes in mathematics, 1405, Springer-Verlag, Heidelberg, (1989), 1-11. Google Scholar

[4]

D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 (1975), 1-25.  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), 97-138. doi: 10.1007/BF00249052.  Google Scholar

[6]

X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation, Mathematical Programming Serie B, 134 (2012), 71-99. 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), 3113-3121. 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), 259-270. 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), 468-511. 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), 806-820. 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), 533-547. 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), 51-68. doi: 10.1080/02331930410001662199.  Google Scholar

[13]

A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737. 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), 493-504. 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), 4047-4058. 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), 61-77. doi: 10.1007/BF01581628.  Google Scholar

[19]

W. I. Zangwill, Nonlinear programing via penalty functions, Management Science, 13 (1967), 344-358.  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), 759-782. 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), 515-525. doi: 10.1080/10556780600627693.  Google Scholar

[3]

A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems, Lecture notes in mathematics, 1405, Springer-Verlag, Heidelberg, (1989), 1-11. Google Scholar

[4]

D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 (1975), 1-25.  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), 97-138. doi: 10.1007/BF00249052.  Google Scholar

[6]

X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation, Mathematical Programming Serie B, 134 (2012), 71-99. 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), 3113-3121. 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), 259-270. 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), 468-511. 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), 806-820. 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), 533-547. 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), 51-68. doi: 10.1080/02331930410001662199.  Google Scholar

[13]

A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737. 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), 493-504. 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), 4047-4058. 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), 61-77. doi: 10.1007/BF01581628.  Google Scholar

[19]

W. I. Zangwill, Nonlinear programing via penalty functions, Management Science, 13 (1967), 344-358.  Google Scholar

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