Study On Fréchet Type Constraint Qualifications Of A Nonconvex Inequality

The basic constraint qualification(BCQ)of convex and strong BCQ inequalities are important concepts in optimization.in approximation theory and mathematical programming these two concepts closely relate to error bounds,the CHIP property,strong CHIP property and(G)propery.Since nonconvex functions appear more frequently and more general in optimization theory and application,it is natural to study constraint qualification of an inequality defined by a nonconvex function.On the basis of BCQ and strong BCQ convex inequality,this paper mainly studies two kinds of constraint qualifications of a nonconvex inequality by Fréchet normal cones and subdifferentials,and these are named as Fréchet BCQ and Fréchet strong BCQ Through the end set of a convex set,this paper proves that Fréchet strong BCQ is equivalent to the Fréchet BCQ and that the end set of Fréchet subdifferential is strictly separated with zero point,this also shows that strong Fréchet BCQ is stronger than Fréchet BCQ.Further,by tangent cone and weak tangent cones of convex sets,this paper gives sufficient and necessary conditions of two kinds of Fréchet type constraint qualifications in the reflexive space and the finite dimensional space,when restricted to convex inequality,two types of constraint qualifications reduce to BCQ and strong BCQ of the convex inequality,and several existing results on BCQ can be also obtained by conclusions given in the paper.By Fréchet normal cone and subdifferential,this paper extends the study of constraints qulificatons from the convex inequality to the nonconvex inequality.