An exact penalty function approach for inequality constrained optimization problems based on a new smoothing technique

Yıl 2023, , 761 - 777, 30.09.2023

Nurullah Yılmaz , Hatice Öğüt

https://doi.org/10.31801/cfsuasmas.1150659

Öz

Exact penalty methods are one of the effective tools to solve nonlinear programming problems with inequality constraints. In this study, a new class of exact penalty functions is defined and a new family of smoothing techniques to exact penalty functions is introduced. Error estimations are presented among the original, non-smooth exact penalty and smoothed exact penalty problems. It is proved that an optimal solution of smoothed penalty problem is an optimal solution of original problem. A smoothing penalty algorithm based on the the new smoothing technique is proposed and the convergence of the algorithm is discussed. Finally, the efficiency of the algorithm on some numerical examples is illustrated.

Anahtar Kelimeler

Constrained optimization, smoothing technique, exact penalty function

Teşekkür

This paper was presented in 4th International Conference on Pure and Applied Mathematics (ICPAM - VAN 2022), Van-Turkey, June 22-23, 2022.

Kaynakça

• Eremin, I. I., The penalty method in convex programming, Cybernetics, 3(4) (1967), 53-56. https://doi.org/10.1007/BF01071708
• Zangwill, W. I., Nonlinear programing via penalty functions, Management Science, 13 (1967), 344-358. http://www.jstor.org/stable/2627851
• Rubinov, A. M., Glover, B. M., Yang, X. Q., Decreasing functions with applications to penalization, SIAM J. Optim., 10 (1999), 289-313. https://doi.org/10.1137/S105262349732609
• Rubinov, A. M., Yang, X. Q., Bagirov, A. M., Penalty functions with a small penalty parameter, Optim. Methods Softw., 17(5) (2002), 931-964. https://doi.org/10.1080/1055678021000066058
• Wu, Z. Y., Bai, F. S., Yang, X. Q., Zhang, L. S., An exact lower-order penalty function and its smoothing in nonlinear programming, Optimization, 53(1) (2004), 51-68. https://doi.org/10.1080/02331930410001662199
• Bai, F. S., Wu, Z. Y., Zhu, D. L., Lower order calmness and exact penalty function, Optim. Methods Softw., 21(4) (2006), 515-525. https://doi.org/10.1080/10556780600627693
• Pinar, M. C., Zenios, S., On smoothing exact penalty functions for convex constrained optimization, SIAM J. Optim., 4(3) (1994), 468-511. https://doi.org/10.1137/0804027
• Chen, C., Mangasarian, O. L., A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 (1996), 97-138. https://doi.org/10.1007/BF00249052
• Bertsekas, D. P., Nondifferentiable optimization via approximation. In: Balinski, M. L., Wolfe, P. (eds) Nondifferentiable Optimization. Mathematical Programming Studies, 3 (1975), 1-25. https://doi.org/10.1007/BFb0120696
• Zang, I., A smoothing out technique for min-max optimization,Math. Programm., (19) 1980, 61-77. https://doi.org/10.1007/BF01581628
• Bagirov, A. M., Al Nuaimat, A., Sultanova, N. Hyperbolic smoothing function method for minimax problems, Optimization, 62(6) (2013), 759-782. https://doi.org/10.1080/02331934.2012.675335
• Yilmaz, N. and Sahiner, A., On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Control Optim., 10(3) (2020), 317-330. https://doi.org/10.3934/naco.2020004
• Wu, Z. Y., Lee, H. W. J., Bai, F. S., Zhang, L. S., Quadratic smoothing approximation to $l_{1}$ exact penalty function in global optimization, J. Ind. Manag. Optim., 1(4) 2005, 533-547. https://doi.org/10.3934/jimo.2005.1.533
• Lian, S. J., Smoothing approximation to l1 exact penalty for inequality constrained optimization, Appl. Math. Comput., 219(6) (2012), 3113-3121. https://doi.org/10.1016/j.amc.2012.09.042
• Xavier, A. E., Hyperbolic penalty: a new method for nonlinear programming with inequalities, Int. Trans. Op. Res., 8(6) (2001), 659-671. https://doi.org/10.1111/1475-3995.t01-1-00330
• Liu, B., On smoothing exact penalty function for nonlinear constrained optimization problem, J. Appl. Math. Comput., 30 (2009), 259-270. https://doi.org/10.1007/s12190-008-0171-z
• Xu, X., Meng, Z., Sun, J., Shen, R., A penalty function method based on smoothing lower order penalty function, J. Comput. Appl. Math., 235(14) (2011), 4047-4058. https://doi.org/10.1016/j.cam.2011.02.031
• Meng, Z., Dang, C., Jiang, M., Shen, R., A smoothing objective penalty function algorithm for inequality constrained optimization problems, Numer. Funct. Anal. Optim., 32(7) (2011), 806-820. https://doi.org/10.1080/01630563.2011.577262
• Sahiner, A., Kapusuz, G., Yilmaz, N., A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Control Optim., 6(2) (2016), 161-173. https://doi.org/10.3934/naco.2016006
• Xu, X., Dang, C., Chan, F., Wang, Y., On smoothing $l_{1}$ exact penalty function for constrained optimization problems, Numer. Funct. Anal. Optim., 40(1) (2019), 1-18. https://doi.org/10.1080/01630563.2018.1483948
• Lee, J. and Skipper, D., Virtuous smoothing for global optimization, J. Glob. Optim., 69 (2017), 677-699. https://doi.org/10.1007/s10898-017-0533-x
• Xu, L., Lee, J. and Skipper, D., More virtuous smoothing, SIAM J. Optim., 29(2) 2019, 1240-1259. https://doi.org/10.1137/18M11728
• Grossmann, C., Smoothing techniques for exact penalty function methods, In: Panaroma of Mathematics: Pure and Applied Contemporary Mathematics,, 658 (2016), 249-265.
• Hassan M., Baharum, A., Generalized logarithmic penalty function method for solving smooth nonlinear programming involving invex functions, Arab. J. Basic Appl. Sci., 26(1) (2019), 202-214. https://doi.org/10.1080/25765299.2019.1600317
• Dolgopolik, M. V., Smooth exact penalty functions: a general approach, Optim. Lett., 10 (2016), 635-648. https://doi.org/10.1007/s11590-015-0886-3
• Lucidi S., Rinaldi, F., Exact penalty functions for nonlinear integer programming problems, J. Optim. Theory Appl., 145 (2010), 479-488. https://doi.org/10.1007/s10957-010-9700-7
• Di Pillo, G., Lucidi, S., Rinaldi, F., An approach to constrained global optimization based on exact penalty functions, J. Glob Optim., 54 (2012), 251-260. https://doi.org/10.1007/s10898-010-9582-0
• Antczak, T., A new exact exponential penalty function method and nonconvex mathematical programming, Appl. Math. Comput., 217(15) (2011), 6652-6662. https://doi.org/10.1016/j.amc.2011.01.051