Generalized Bishop-Phelps Cone And Its Application

Optimization theory is a branch of applied mathematics, which is used widely in real life. Optimization become an independent discipline during1930s and1940s, but optimization idea has appeared at the beginning of the creation of calculus. A very well-known simplex method is proposed by the Dantizig of the RAND Company in1947, which is applicable to all linear optimization problem of solving; In1979a new polynomial time algorithm is presented by the Soviet mathematician Khachiyan, namely the ellipsoid method, which is better than the simplex method in theory firstly; With the widely application of the optimization theory in the aspect of economic planning, production management and transportation. Several branches of optimization theory has get greatly advancement and depthly study, such as Nonlinear programming, multi-objective programming, non-smooth optimization and integer programming; The cone and its relevant theories are the powerful means for studying optimization theory and optimization methods, recently its research has been focused by domestic and foreign scholars. At the same time, the quotient space has also become the focus of research as an important definition of functional analysis.The main contents of this paper as follows:Firstly, based on the definition of the Bishop-Phelps cone, Nuclear cone and Full Nuclear cone, the definition of p cone is taken, and some properties of generalized Bishop-Phelps cone are discussed, the properties and theorems are studied and discussed in the general vector space, and its detail proof are given on the base of the properties of the Bishop-Phelps cone in real normed vector space.Secondly, based on the given concept of Quotient-space and definition of convex cone. The concept of cone pseudo quotient space is taken, its properties are studied.In conclude, the definition of p cone is given by spreading the concept of the Bishop-Phelps cone, its several properties are studied in the vector space and locally convex space, which advanced research provides theoretical support for the deeper study of Pareto validity and Ekeland variational principle. At the same time, the study of the cone pseudo quotient space could play an important role in the research of convex cone.