The Regularity Of Generalized Convex Sets In Normed Linear Spaces

Optimization theory is not only the most important part of optimization, but also an important theoretical basis in operations research. As a fundamental theory of optimization theory, the regular property is a crucial point for the stability and sensitivity analysis in feasibility problems in the optimization. The GPR property of a family of closed convex sets with nonempty intersection was introduced by Gurin, Poliac and Raik in 1967, which is in fact equivalent to the regularity property of a finite family of closed convex sets with nonempty intersection introduced by Bauschke and Borwein. In recent decades, the tools of researching the Optimization theory, Set-valued analysis and Variational analysis have been rapidly developed, thereby which promote the research of linear regularity and metric regularity and obtain some practical conclusions, and make more general regularity to be focus of domestic and foreign scholars.This article can be summarized in two parts:In the first part, by using of set-valued mapping, Proximinal set, core and Robinson-Ursescu theorem, this paper discusses the regularity condition of finitely many family of closed convex sets with nonempty intersection in Frechet space, it is obtained that the family satisfies regularity condition when there exists i∈I such that Ci is Proximinal, the intersection of the family is bounded and its core is nonempty. On the other hand, we conclude that the regularity still holds when the strong quasi relative interior of the family is nonempty and its intersection is bounded in Banach space. At the same time, we also analyze that the polyhedron whose relative interior is nonempty and bounded also exists regularity. Those are from a side answered Stefan Maruster and Cristina Popirlan's problem whether a family whose intersection is bounded but interior is empty can satisfy regularity condition.In the second part, the paper introduces pseudo-convex sets and quasi-convex sets, and gets the operation properties of pseudo-(quasi-, strong quasi-)relative interior of pseudo-convex in normed linear space, but also studies a regularity condition of pseudo-convex sets if the intersection of family is equal to the intersection of its convex hull, and obtains the regular condition only if its intersection is bounded and interior is nonempty. Finally, using the property of quasi-convex and combining with the regularity condition of closed convex sets, it discusses conditions of the family in Frechet space and Banach space.Beginning of the regularity property of a finite family of closed convex sets, and introducing pseudo-convex and quasi-convex, the paper makes a preliminary study on regularity conditions of generalized convex sets and obtains the corresponding results, which would provide some theoretical supporting for the stability and sensitivity analysis of the general convex feasibility problems.