On the convergence of a smooth penalty algorithm without computing global solutions. (English) Zbl 1235.90120

Summary: We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problems.


90C26 Nonconvex programming, global optimization
65K05 Numerical mathematical programming methods
90C25 Convex programming
Full Text: DOI


[1] W. I. Zangwill, “Non-linear programming via penalty functions,” Management Science, vol. 13, pp. 344-358, 1967. · Zbl 0171.18202
[2] A. Auslender, R. Cominetti, and M. Haddou, “Asymptotic analysis for penalty and barrier methods in convex and linear programming,” Mathematics of Operations Research, vol. 22, no. 1, pp. 43-62, 1997. · Zbl 0872.90067
[3] A. Ben-Tal and M. Teboulle, “A smoothing technique for non-differentiable optimization problems,” in Optimization, vol. 1405 of Lecture Notes in Mathematics, pp. 1-11, Springer, Berlin, Germany, 1989. · Zbl 0683.90078
[4] C. H. Chen and O. L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems,” Mathematical Programming, vol. 71, no. 1, pp. 51-69, 1995. · Zbl 0855.90124
[5] C. Chen and O. L. Mangasarian, “A class of smoothing functions for nonlinear and mixed complementarity problems,” Computational Optimization and Applications, vol. 5, no. 2, pp. 97-138, 1996. · Zbl 0859.90112
[6] M. Herty, A. Klar, A. K. Singh, and P. Spellucci, “Smoothed penalty algorithms for optimization of nonlinear models,” Computational Optimization and Applications, vol. 37, no. 2, pp. 157-176, 2007. · Zbl 1181.90222
[7] M. \cC. Pinar and S. A. Zenios, “On smoothing exact penalty functions for convex constrained optimization,” SIAM Journal on Optimization, vol. 4, no. 3, pp. 486-511, 1994. · Zbl 0819.90072
[8] I. Zang, “A smoothing-out technique for min-max optimization,” Mathematical Programming, vol. 19, no. 1, pp. 61-77, 1980. · Zbl 0468.90064
[9] C. C. Gonzaga and R. A. Castillo, “A nonlinear programming algorithm based on non-coercive penalty functions,” Mathematical Programming, vol. 96, no. 1, pp. 87-101, 2003. · Zbl 1023.90061
[10] Z. Y. Wu, F. S. Bai, X. Q. Yang, and L. S. Zhang, “An exact lower order penalty function and its smoothing in nonlinear programming,” Optimization, vol. 53, no. 1, pp. 51-68, 2004. · Zbl 1079.90130
[11] Z. Meng, C. Dang, and X. Yang, “On the smoothing of the square-root exact penalty function for inequality constrained optimization,” Computational Optimization and Applications, vol. 35, no. 3, pp. 375-398, 2006. · Zbl 1128.90055
[12] O. L. Mangasarian and S. Fromovitz, “The Fritz John necessary optimality conditions in the presence of equality and inequality constraints,” Journal of Mathematical Analysis and Applications, vol. 17, pp. 37-47, 1967. · Zbl 0149.16701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.