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.

[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.

[3]

A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems,, Lecture notes in mathematics, (1989), 1.

[4]

D. Bertsekas, Nondifferentiable optimization via approximation,, Mathematical Programming Study, 3 (1975), 1.

[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.

[6]

X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation,, Mathematical Programming Serie B, 134 (2012), 71. 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. 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. 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. 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. 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. 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. doi: 10.1080/02331930410001662199.

[13]

A. E. Xavier, The hyperbolic smoothing clustering method,, Pattern Recognition, 43 (2010), 731.

[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.

[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.

[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. doi: 10.1007/BF01581628.

[19]

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

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.

[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.

[3]

A. Ben-Tal and M. Teboule, Smoothing technique for nondifferentiable optimization problems,, Lecture notes in mathematics, (1989), 1.

[4]

D. Bertsekas, Nondifferentiable optimization via approximation,, Mathematical Programming Study, 3 (1975), 1.

[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.

[6]

X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation,, Mathematical Programming Serie B, 134 (2012), 71. 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. 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. 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. 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. 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. 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. doi: 10.1080/02331930410001662199.

[13]

A. E. Xavier, The hyperbolic smoothing clustering method,, Pattern Recognition, 43 (2010), 731.

[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.

[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.

[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. doi: 10.1007/BF01581628.

[19]

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

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