On Clarke Constraint Qualifications Of A Lipschitz Inequality And Global Error Bounds

Constraint qualifications are important concepts in mathematical optimization and mathematical programming.In many related works of scholars at home and abroad,the constraint qualification is involved.For example,constraint qualifications were used to study Fenchel duality and the formula of subdifferentials of convex functions.And constraint qualifications involving epigraphs were applied to the extended Farkas lemma and Lagrange duality in convex programming.Besides,constraint qualifications were widely applied in the study of two kinds of conditional optimization problems with different convex functions.In addition,the constraint specification of the closed cone is applied to the Lagrange-dual programming in the quasi-convex case.Among these studies,the most widely used is the basic constraint qualification.There are many scholars who have studied BCQ and strong BCQ of a convex inequality.In general,the strong BCQ implies the BCQ,but the converse implication fails.When the function is the maximum of finitely many smooth convex functions,it also has been proved that BCQ of the convex inequality is equivalent to the strong BCQ.In these studies,strong BCQ and BCQ are restricted to the convex inequality by a continuous convex function.However,it is more general that non-convex functions appear in the study of theory and application;and thus it is natural to study constraint qualifications of the non-convex inequality.Based on BCQ and strong BCQ of the convex inequality,this paper is to study constraint qualifications of the non-convex inequality defined by a Lipschitz function(not convex necessarily)and use Clarke normal cone and subdifferentials to discuss two types of constraint qualifications that are named as Clarke BCQ and Clarke strong BCQ.The relation between these two BCQs is investigated and several sufficient and/or necessary conditions ensuring two BCQs are also provided.As applications,we studied global error bounds of the convex inequality defined by the Clarke directional derivative of the Lipschitz function.It is proved that Clarke strong BCQ is satisfied if and only if Clarke BCQ holds and the convex inequality has the global error bounds.