Journal of Applied Mathematics
Volume 2012 (2012), Article ID 691018, 19 pages
Research Article

Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

1College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China
2Research Institute of Information and System Computation Science, Beifang University of Nationalities, Yinchuan 750021, China

Received 8 February 2012; Accepted 1 April 2012

Academic Editor: C. Conca

Copyright © 2012 Qilin Wang and Guolin Yu. 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.


The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.