Finally, the efficiency of the algorithm on some numerical examples is illustrated.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:Exact penalty methods are one of the effective tools to solve nonlinear programming problems with inequality constraints.
📊
Key Findings & Data: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.
💡
Conclusion & Impact:Finally, the efficiency of the algorithm on some numerical examples is illustrated.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Constrained optimization · smoothing technique · exact penalty function
Abstract
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.
YILMAZ, N., ÖĞÜT, H. (2023). An exact penalty function approach for inequality constrained optimization problems based on a new smoothing technique. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Cilt: 72 Sayı: 3, 761–777. https://ajindex.com/articles/964136
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
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.2012.0