On lower bounds of the second-order directional derivatives of Ben-Tal, Zowe, and Chaney

• 成果类型: Article
• 署名作者: Huang, LR; Ng, KF
• 署名单位: Chinese University of Hong Kong
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.22.3.747
• 发表日期: 1997
• 页码: 747-753
• 关键词: nonsmooth optimization; optimality conditions; epi-derivatives; 2ND-ORDER

摘要

Let f be a regular, locally Lipschitz real-valued function defined on an open convex subset of a normed space. We show that at any unit direction u, the upper second-order derivative D(+)(2)f(.;u, 0) (in the sense of Dem'yanov and Pevnyi 1974; Ben-Tal and Zowe 1982) has the same lower bounds as the lower second-order derivatives D(-)(2)f(.; u, 0). Consequently, one can characterize the convexity of f in terms of these derivatives. We also obtain the corresponding results in terms of Chaney's second-order directional derivatives.