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

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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

Abstract
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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 and 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.
[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.
[3]	A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems, Lecture notes in mathematics, 1405, Springer-Verlag, Heidelberg, (1989), 1-11.
[4]	D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 (1975), 1-25.
[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.
[6]	X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation, Mathematical Programming Serie B, 134 (2012), 71-99. doi: 10.1007/s10107-012-0569-0.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737.
[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.
[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.
[16]	N. Yilmaz, A. Sahiner, A New Global Optimization Technique Based on the Smoothing Approach for Non-smooth, Non-convex Optimization,, Submitted., ().
[17]	N. Yilmaz, A. Sahiner, Smoothing Approach for Non-lipschitz Optimization,, Submitted., ().
[18]	I. Zang, A smooting out technique for min-max optimization, Mathematical Programming, 19 (1980), 61-77. doi: 10.1007/BF01581628.
[19]	W. I. Zangwill, Nonlinear programing via penalty functions, Management Science, 13 (1967), 344-358.

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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.
[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.
[3]	A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems, Lecture notes in mathematics, 1405, Springer-Verlag, Heidelberg, (1989), 1-11.
[4]	D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 (1975), 1-25.
[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.
[6]	X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation, Mathematical Programming Serie B, 134 (2012), 71-99. doi: 10.1007/s10107-012-0569-0.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 (2010), 731-737.
[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.
[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.
[16]	N. Yilmaz, A. Sahiner, A New Global Optimization Technique Based on the Smoothing Approach for Non-smooth, Non-convex Optimization,, Submitted., ().
[17]	N. Yilmaz, A. Sahiner, Smoothing Approach for Non-lipschitz Optimization,, Submitted., ().
[18]	I. Zang, A smooting out technique for min-max optimization, Mathematical Programming, 19 (1980), 61-77. doi: 10.1007/BF01581628.
[19]	W. I. Zangwill, Nonlinear programing via penalty functions, Management Science, 13 (1967), 344-358.

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