Study On Abadie CQ And Strong Abadie CQ Of Convex Constraint Systems

Convex optimization is an important topic and has been studied by many authors.Convex optimization has been widely used in many fields.The classic differentiable convex optimization problem is to establish the optimality condition of the solution.Fritz John first proposed the necessary optimality conditions.For the convex programming optimization problem,Kuhn and Tucker provided the necessary and sufficient conditions for the optimal solution of the optimization problem,and this optimization condition is well known for Karush-Kuhn-Tucker(KKT)condition.In the optimization theory,the KKT condition is also called the Kuhn-Tucker condition.Note that the Abadie constraint qualifications(ACQ)is the weakest condition to ensure that the KKT condition characterizations for the optimal solution,this paper mainly study the ACQ of the convex constraint system.This paper mainly considers a convex constraint system defined by a closed convex multifunction and a closed convex subset.By reviewing the ACQ of the convex inequality and the ACQ of a closed convex set-valued mapping,definitions of ACQ and strong ACQ of a convex constraint system is given.This ACQ can cover two cases of the classical convex inequality and closed convex set-valued mapping.Then,some sufficient conditions and necessary conditions for ACQ and strong ACQ of the convex constraint system are given in this paper,and we also proved that the strong ACQ of the convex constraint system implies ACQ,but the converse inclusion may not be true.Using the normal cone,the co-derivative,the tangent cone and the tangent derivative,we obtain the equivalent conditions of the convex constraint system ACQ and strong ACQ.In particular,this paper proves the exact quantitative relation between strong ACQ and its equivalent conditions on coefficients.