(F,K,B)-Vex Sets And Z-Convex Function And Some Related Properties

The theory of optimizaiton is a widely used discipline, it discusses the characteristics of best choice for decision-making, Structure calculation method to seek the optimal solution, research these calculation method theories and the actual calculated performance. In recent years, research on the theory of optimization-set-valued analysis, nonlinear analysis and other branches obtained considerable development, and the actual applied more and more widelyIn nonlinear analysis and set-valued analysis, convexity plays an important role. In the past few decades, various extensions of convexity have been given, enriches the contents of nonlinear analysis. In1999, Youness introduced the concept of E-convex sets. Then, some extensions of the E-convex set were proposed. Due to the applications of generalized convexity and set-valued analysis in image processing, computational geometry, error estimation and computer science, we are inspired to investigate generalized convexity from the angles of set-valued analysis. In addition, some special generalized convex function are involved in many areas of applied mathematics, such as Godunova-Levin function, s-convex function, P function. In2007, Varoanec gave a new kind of funtions—h-convex functions which generalize all the functions mentioned above and the properties of these function have been investigated by many authors.In the first part, the concept of E-b-convex set given by Mishra, Mohapatra and Youness is extended to (F, K, b)-convex set and some specific examples are given. The properties about the intersection and sum of (F, K, b)-convex set in vector spaces are discussed and some sufficient conditions for these properties are given.In the second part, based on the research of the S-convex function, P function, Godunova-Levion function, the concept of Z-convex function is given, and some properties about funtion operation (addition, scalar multiple, composite etc.) are discussed.