Properties Of A Class Of Abstract Convex And Its Application In Vector 0ptimization Problems

The vector optimization problem is to maximize or minimize the vector-valued function under certain conditions.The research on this problem involves non-smooth analysis,convex analysis,functional analysis and other disciplines,which has attracted many scholars.Co-radiant sets and scalar functions are important tools for the study of vector optimization,among which co-radiant sets is the basic tool for the study of unified solutions of vector optimization problems.This paper first studies these two special sets in the framework of abstract convexity : The properties of radiant sets and co-radiant sets,using Minkowski functionals to give the equivalent characterization of radiant sets and co-radiant sets,thus establishing the characterization of the unified solution set of the vector optimization problem.Then,Under the co-radiant sets,the abstract convexity of a special class of nonlinear scalar functions is studied,and this result is applied to the nonlinear scalar theory of unified solutions for vector optimization problems.The main contents are arranged as follows:In the first part,based on the abstract convex theory,using the Minkowski gauge function and the Minkowski co-gauge function,the equivalent characterizations of two special abstract convex(concave)sets are given,and they are applied to the characterization of the unified solution set of vector optimization.First of all,using the abstract convex theory and the Minkowski functional,it is proved that the radiant set and the co-radiant set can be described equivalently with positive homogeneous functions,so that they can be derived with the help of positive homogeneous functions.Important properties,and give an example to illustrate the rationality of the relevant results.Finally,we apply the important properties to the approximate solution of the vector optimization problem,and obtain a characterization of the approximate solution set of the co-radiant set.In the second part,consider the introduction of a nonlinear scalar function ? in the sense of abstract convexity,studies some of its properties as an abstract convex function,so as to give the scalarization result of the vector optimization problem.First,The relationship between ? and its positive homogeneous expansion function is studied,and the nature of the positive homogeneous function is used to describe the abstract subdifferential when ? is an ICR function.Then,We found that its positive homogeneous extension function satisfies convexity under certain conditions,and thus studied the subdifferential under the convex meaning of its positive homogeneous extension function.Finally,established a positive homogeneous extension function based on these properties characterization of approximate solution of scalarization problem.