Journal of Applied Mathematics
Volume 2012 (2012), Article ID 620949, 12 pages
Research Article

On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions

1School of Science, Shandong University of Technology, Zibo 255049, China
2Institute of Operations Research, Qufu Normal University, Qufu 273165, China

Received 18 September 2011; Accepted 9 November 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Bingzhuang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 problem.