Study On Characterizations For Two Types Of Constraint Qualifications Via Singular Subdifferential

Constraint qualifications constitute an important part of the necessary optimali-ty conditions,and have been widely used in optimization and mathematical program-ming,such as KKT optimality conditions,sensitivity analysis of optimization problems in Hilbert spaces,the description of the tangent cones to a feasible subset,and error bounds of the inequality.Currently,many types of constraint qualifications have been studied and developed.One of the most important constraint qualifications is the Basic Constraint Qualification(BCQ).Many scholars have studied this kind of constraint qualifications,and proved.characterizations for BCQ and strong BCQ.Generally,strong BCQ implies BCQ,but the converse is not necessarily true.Note that,BCQ and strong BCQ are studied in the context of the convex inequal-ity defined by a continuous convex function.Non-convex and discontinuous functions are more general and appear more frequently in theory and applications.Therefore,a natural issue is to study constraint qualifications of the inequality by a non-convex and discontinuous function.In this paper,we mainly study constraint qualifications and its characterizations of inequality by semi-continuous and non-convex function which is the generalization of BCQ and strong BCQ in convex case.One is the extended Clarke BCQ and the extended Clarke strong BCQ with Clarke singular subdifferential.The equiva-lent characterization of these two constraint qualifications is discussed.The other is the extended Frechet BCQ and the extended Frechet strong BCQ with Frechet singular sub-differential.And the equivalent characterization of these two constraint qualifications is also discussed.Furthermore,we prove that when the singular subdifferential is trivial(i.e.containing only zero elements),these two kinds of constraint qualifications studied above are equivalent to the Clarke constraint qualifications and Frechet constraint qualifications discussed in[25],which further extends the study of constraint qualifications.