Bezier Curve Based Smoothing Penalty Function for Constrained Optimization

Year 2020, Volume: 3 Issue: 2, 70 - 79, 25.03.2021

Ahmet Sahiner , Nurullah Yılmaz , Gülden Kapusuz , Gamze Özkardaş

Abstract

In this study, we consider nonlinear inequality constrained optimization
problems. We introduce l1 exact penalty function approach with a new smoothing
function based on Bezier curve. Then, we propose a new algorithm by using the differentiation
based methods to solve for solving l1 exact penalty functions. Finally,
we apply our algorithm to test problems to demonstrate the effectiveness of the algorithm

Keywords

l1 penalty function, Smoothing, Non-smooth optimization

Supporting Institution

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Project Number

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Thanks

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References

• W. I. Zangwill, Nonlinear programing via penalty functions, Manage Sci., 13 1967, 344–358.
• M. V. Dolgopolik, Smooth exact penalty functions: a general approach, Optim. Lett. 10(3) 2016,635–648.
• M. V. Dolgopolik, Smooth exact penalty functions II: a reduction to standard exact penalty functions, Optim. Lett., 10(7) 2016, 1541–1560.
• J.Liu, R. Ma, X. Zeng, W. Liu, M. Wang, H. Chen, An efficient non-convex total variation approach for image deblurring and denoising, Appl. Math. Comput., 397 2021, 125977.
• C. Ma, X. Li, K. F. Cedric Yiu, L.-S Zhang, New exact penalty function for solving constrained finite min-max problems, Appl. Math. Mech., 33(2) 2012, 253–270 .
• A. Jayswal, An exact l1 penalty function method for multi-dimensional first-order PDE constrained control optimization problem, Eur. J. Control, 52 2020, 34-41.
• R. Manikantan, S. Chakraborty, T. K. Uchida, C. P. Vyasarayani, Parameter Identification in Nonlinear Mechanical Systems with Noisy Partial State Measurement Using PID-Controller Penalty Functions Mathematics 8 2020, 1084.
• D. Bertsekas, Nondifferentiable optimization via approximation, Mathematical Programming Study, 3 1975, 1–25.
• I. Zang, A smooting out technique for min-max optimization, Math. Programm., 19 1980, 61–77.
• A. Ben-Tal, M. Teboule, Smoothing technique for nondifferentiable optimization problems, Lecture notes in mathematics, 1405, Springer-Verlag, Heidelberg, 1989, 1-11.
• C. Chen, O.L. Mangasarian, A Class of Smoothing Functions for Nonlinear and Mixed Complementarity Problem, Comput. Optim. Appl., 5 1996, 97–138.
• A. M. Bagirov, A. Al Nuamiat, N. Sultanova Hyperbolic smoothing functions for nonsmooth minimization, Optimization, 62 (6), 2013, 759–782.
• A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recognition, 43 2010, 731–737.
• A. E. Xavier,A. A. F. D. Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Glob. Optim. 31 (2005) 493-504.
• C. Grossmann, Smoothing techniques for exact penalty function methods, Contemporary Mathematics, In book Panaroma of Mathematics: Pure and Applied, 658 249–265.
• N. Yilmaz, A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numer. Funct. Anal. Optim., 40(5) 2019, 523–534.
• N. Yilmaz, A. Sahiner, On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Control Optim., 10 (3) 2020, 317–330.
• M.C. Pinar, S. Zenios, On smoothing exact penalty functions for convex constrained optimization, SIAM J. Optim., 4 1994, 468-511.
• S. J. Lian, Smoothing approximation to l1 exact penalty for inequality constrained optimization, Appl. Math. Comput., 219 2012, 3113–3121.
• B. Liu, On smoothing exact penalty function for nonlinear constrained optimization problem, J. Appl. Math. Comput., 30 2009, 259–270.
• X. Xu, Z. Meng, J. Sun, R. Shen A penalty function method based on smoothing lower order penalty function, J. Comput. Appl. Math., 235 2011, 4047-4058.
• Z. Meng, C. Dang, M. Jiang, R. Shen A smoothing objective penalty function algorithm foe inequality constrained optimization problems, Numer. Funct. Anal. Optim., 32 2011, 806-820.
• Z. Y.Wu, H.W. J. Lee, F. S. Bai, L. S. Zhang, Quadratic smoothing approximation to l1 exact penalty function in global optimization, J. Ind. Manage. Optim., 53 2005, 533–547.
• Z.Y.Wu, F.S.Bai, X.Q.Yang, L.S.Zhang, An exact lower orderpenalty function and its smoothing in nonlinear programming, Optimization, 53 2004, 51-68.
• F.S.Bai, Z.Y.Wu, D.L. Zhu, Lower order calmness and exact penalty fucntion, Optim. Methods Softw., 21 2006, 515–525.
• A. Sahiner, G. Kapusuz, N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Control Optim., 6 (2) 2016, 161–173.
• X. Xu, C. Dang, F. T. S. Chan and Yongli Wang, On smoothing l1 exact penalty function for constrained optimization problems Numer. Funct. Anal. Optim., 40 (1) 2019, 1–18
• J. Min, Z. Meng, G. Zhou, R. Shen, On the smoothing of the norm objective penalty function for two-cardinality sparse constrained optimization problems, Neurocomputing, 2020 Doi: 10.1016/j.neucom.2019.09.119.
• Qian Liu, Yuqing Xu, Yang Zhou, A class of exact penalty functions and penalty algorithms for nonsmooth constrained optimization problems, J. Glob. Optim., 76 2020, 745–768.
• X. Chen, Smoothing Methods for nonsmooth, nonconvex minimzation, Math. Programm. Serie B, 134 2012, 71–99.
• A. Sahiner, N. Yilmaz and G. Kapusuz, A new global optimization method and applications, Carpathian Math. J., 33(3) 2017, 373-380.
• G.E. Farin, Curves and Surfaces for CADG: A Partical Guide, Morgan Kaufmann, San Fransico, 2002.