Calmness And Error Bounds Of Multifunctions

In contrast to many existing results on calmness,error bounds and their stability properties established via dual notions such as normal cone,subdifferential,coderivative and so on,this thesis is devoted to conducting research on primal sufficient and/or necessary conditions for the calmness and error bounds.First,we mainly consider several primal sufficient and/or necessary conditions for calmness of nonconvex multifunctions and nonconvex generalized equations with constraints.In terms of the Bouligand tangent cone and Clarke tangent cone,we introduce and study the Abadie CQ and the strong ACQ for a nonconvex multifunction.Using the strong ACQ,we establish several primal sufficient and/or necessary conditions for calmness of a multifunction with the Shapiro property（an extension of both convexity and smoothness）.With the help of tangent cones and tangent derivatives,we also consider sufficient and/or necessary conditions for calmness of generalized equations with constraints under the assumption that the objective multifunction is generalized-differentiable and that the constraint set has the Shapiro property.Our results improve and extend some existing ones from the convex case to the nonconvex case.Second,in terms of the Slater condition of the Bouligand and Clarke tangent derivatives of the objective multifunction Φ,we mainly study the stability of error bound of Φ at a point x with respect to an ordering cone K.We prove that the Slater condition of the Bouligand tangent derivative of Φ at x with respect to K is always stable with respect to all small calm perturbations.Based on this result,we prove that the Slater condition of the Bouligand tangent derivative of Φ at x with respect to K is a sufficient condition for Φ to have a stable error bound at x with respect to K when Φundergoes small calm and regular perturbations.Moreover,as applications,we consider a linear regularity property for a finite or an infinite collection of closed（not necessarily convex）sets and establish some primal sufficient and/or necessary conditions for this property,which extend and improve some existing results to either the nonconvex case or the case with an infinite index set.We also provide sufficient conditions for a convex progress to have a stable global error bound with respect to an ordering cone.