Study On Characterizations Of Basic Constriant Qualifications For Gener-Alized Convex Equations

The basic constraint qualifications of generalized convex equations play an important role in optimization theory and mathematical equations.It contains a lot of optimization problems.For example,constraint qualifications were used to study Fenchel duality and the formula of subdifferentials of convex functions.One Farkas-type constraint qualifi-cation has been proved to be useful in convex programming and DCdifference of two convex functions)programming.Constraint qualifications involving epigraphs were ap-plied to the extended Farkas lemma and Lagrange duality in the convex programming.Constraint qualifications were also applied to study of optimality conditions in convex and DC optimization problems.Recently one type of closed cone constraint qualifications were used to study Lagrange duality in quasi-convex programming.In this paper,we study a class of generalized convex equations defined by closed convex multivalued mappings and closed convex constrained sets in Banach space,mainly discussing the characteristics of basic constraint specifications BCQ and strong BCQ and their applications.Using the properties of the convex set of the convex set,we prove that the generalized convex equation has the characteristics of strong basic constraint specifications,and gives the difference between strong BCQ and BCQ.As an application,we apply the characteristics to describe metric subregularity of the generalized convex equations.In the first part.,we study the characteristics of the strong BCQ of generalized convex equations.By the end set of the convex set,we prove that the generalized convex equation has strong BCQ if and only if the generalized convex equations is BCQ and the distance from the zero point to the corresponding set is constant greater than zero.Furthermore,using the tangent cone and the co-derivative,we prove the original feature of the tan-gent cone of strong BCQ and establish the exact coefficient relationship between the strong BCQ and the original feature.In the second part,we study the characteristics of metric subregularity for the gener-alized convex equations.Using the dual features of metric subregularity and the charac-teristics of the strong BCQ,and we prove the original features of metric subregularity for the generalized convex equations.An accurate quantitative relationship between metric subregularity and the original features is established.