A new perturbed smooth penalty function for inequality constrained optimization

Volume 23, Issue 1, pp 36--48 http://dx.doi.org/10.22436/jmcs.023.01.04

Publication Date: October 02, 2020 Submission Date: December 05, 2019 Revision Date: January 24, 2020 Accteptance Date: April 22, 2020

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1038 Downloads
• 2207 Views

Authors

Touna Yang - Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China. Nguyen Thanh Binh - Department of Education and Training Yen Bai, Yen Bai city 32000, Vietnam. Truong Minh Chinh - Viet Tri University of Industry, Viet Tri city 35000, Vietnam. Le Huu Tung - Viet Tri University of Industry, Viet Tri city 35000, Vietnam. Duong Thi Hoa - Yen Bai Teacher's Training College, Yen Bai city 32000, Vietnam.

Abstract

In this paper, we first propose a new smoothing approach to the nonsmooth penalty function for constrained optimization problems, and then show that an approximate optimal solution of the original problem can be obtained by solving an optimal solution of the smoothed optimization problem. Based on the perturbed smooth exact penalty function, we develop an algorithm respectively to finding an approximate optimal solution of the original constrained optimization problem and prove the convergence of the proposed algorithm. The effectiveness of the smoothed penalty function is illustrated through three examples, which show that the algorithm seems efficient.

Share and Cite

Keywords

• Exact penalty function
• inequality constrained optimization
• smoothing method
• optimal solution

MSC

•  90C30
•  65K05
•  57R12

References

• [1] I. Ahmad, A. Ayswal, J. Banerjee, On interval-valued optimization problems with generalized invex functions, J. Inequal. Appl., 2013 (2013), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] M. S. Bazaraa, J. J. Goode, Sufficient conditions for a globally exact penalty function without convexity, Math. Programming Stud., 19 (1982), 1--15
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] N. T. Binh, Smoothing approximation to $l_1$ exact penalty function for constrained optimization problems, J. Appl. Math. Inform., 33 (2015), 387--399
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] N. T. Binh, Smoothed lower order penalty function for constrained optimization problems, IAENG Int. J. Appl. Math., 46 (2016), 76--81
  • View Article
  • MathSciNet
  • Google Scholar


• [5] N. T. Binh, Y. Q. Bai, X. Yan, T. Yang, Perturbed smoothing approach to the lower order exact penalty functions for nonlinear inequality constrained optimization, Tamkang J. Math., 50 (2019), 37--60
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] C. H. Chen, O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Math. Programming, 71 (1995), 51--69
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] G. Di Pillo, L. Grippo, An exact penalty function method with global convergence properties for nonlinear programming problems, Math. Programming, 36 (1986), 1--18
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] J. B. Etoa Etoa, Solving quadratic convex bilevel programming problems using a smoothing method, Appl. Math. Comput., 217 (2011), 6680--6690
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] S. P. Han, O. L. Mangasrian, Exact penalty function in nonlinear programming, Math. Programming, 17 (1979), 251--269
  • View Article
  • MathSciNet
  • Google Scholar


• [10] J. B. Lasserre, A globally convergent algorithm for exact penalty functions, European J. Oper. Res., 7 (1981), 389--395
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] H. Li, L. Zhang, Y.-C. Jiao, An interactive approach based on a discrete differential evolution algorithm for a class of integer bilevel programming problems, Internat. J. Systems Sci., 47 (2016), 2330--2341
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] Z. Q. Meng, C. Y. Dang, X. Q. Yang, On the smoothing of the square-root exact penalty function for inequality constrained optimization, Comput. Optim. Appl., 35 (2006), 375--398
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] J. Nocedal, S. J. Wright, Numerical Optimization, Springer-Verlag, New York (1999)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] M. C. Pinar, S. A. Zenios, On smoothing exact penalty function for convex constrained optimization, SIAM J. Optim., 4 (1994), 486--511
  • View Article
  • MathSciNet
  • Google Scholar


• [15] L. Q. Qi, X. J. Tong, Y. J. Wang, Computing power system parameters to maximize the small signal stability margin based on min-max models, Optim. Eng., 10 (2009), 465--476
  • View Article
  • Google Scholar
  • MATH


• [16] E. Rosenberg, Exact penalty functions and stability in locally Lipschitz programming, Math. Programming, 30 (1984), 340--356
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] M. Ruiz Galán, Elementary convex techniques for equilibrium, minimax and variational problems, Optim. Lett., 12 (2018), 137--154
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] Y. H. Sun, L. S. Wang, Optimality conditions and duality in nondifferentiable interval-valued programming, J. Ind. Manag. Optim., 9 (2013), 131--142
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] C. W. Wang, Y. J. Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. Glob. Optim., 44 (2009), 283--296
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Z. Y. Wu, F. S. Bai, X. Q. Yang, L. S. Zhang, An exact lower order penalty function and its smoothing in nonlinear programming, Optimization, 53 (2004), 51--68
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] Z. Y. Wu, H. W. J. Lee, F. S. Bai, L. S. Zhang, Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization, J. Ind. Manag. Optim., 1 (2005), 533--547
  • View Article
  • MathSciNet
  • Google Scholar


• [22] T. Yang, N. T. Binh, T. M. Thang, D. T. Hoa, A new smoothing nonlinear penalty function for constrained optimization, Math. Comput. Appl., 22 (2017), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [23] C. J. Yu, K. L. Teo, L. S. Zhang, Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895--910
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] W. I. Zangwill, Non-linear programming via penalty functions, Management Sci., 13 (1967), 344--358
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] S. A. Zenios, M. C. Pinar, R. S. Dembo, A smooth penalty function algorithm for network-structured problems, Eur. J. Oper. Res., 83 (1995), 220--236
  • View Article
  • MathSciNet
  • Google Scholar