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Mathematical Problems in Engineering
Volume 2014, Article ID 158780, 10 pages
Research Article

Error Bounds and Finite Termination for Constrained Optimization Problems

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 6 February 2014; Accepted 3 April 2014; Published 30 April 2014

Academic Editor: Changzhi Wu

Copyright © 2014 Wenling Zhao 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 present a global error bound for the projected gradient of nonconvex constrained optimization problems and a local error bound for the distance from a feasible solution to the optimal solution set of convex constrained optimization problems, by using the merit function involved in the sequential quadratic programming (SQP) method. For the solution sets (stationary points set and points set) of nonconvex constrained optimization problems, we establish the definitions of generalized nondegeneration and generalized weak sharp minima. Based on the above, the necessary and sufficient conditions for a feasible solution of the nonconvex constrained optimization problems to terminate finitely at the two solutions are given, respectively. Accordingly, the results in this paper improve and popularize existing results known in the literature. Further, we utilize the global error bound for the projected gradient with the merit function being computed easily to describe these necessary and sufficient conditions.