The Study Of The Differentiability/Sub-differentiability Of Interval-value Mapping And Its Application

Since R.E.Moore gave the operating theory of interval numbers systematically,with the joint efforts of many scholars,interval analysis and its applications have been greatly developed.Interval-value mapping is a function of interval number,which is an important part of interval analysis.The concept of differentiability and related properties of interval-value mapping play an important role in the study of interval-value optimization theory and interval-value differential equation theory.There are two ways to define the differentiability of interval-value mapping:H-differentiability is given by H-difference and g H-differentiability given by g H-difference.In order to better discuss the differentiability and application of interval-value mapping,a new concept of differentiability(D-differentiability)of interval value-mapping is established by using the idea of total differential of real-value function.Meanwhile,we study the concept of interval-value mapping with weaker analytical property than differentiability,and establish the concept of interval-value mapping with sub-differentiability.The specific research work is as follows:1.Firstly,we give the concepts of bounded and definite bounds of subsets on the interval number space,and give the existence theorem of definite bounds.Then we discuss the semi-continuity of interval-valued mapping,and give the concept of semi-continuity of interval-valued mapping and related properties.On this basis,we discuss the convexity of semi-continuous interval-value mapping,and give two sufficient conditions for semi-continuous interval-value mapping to be convex interval-value mapping.2.Firstly,we discuss the differentiability of interval-value mapping by using the thought method of full differentiation of real-value function,establish the concept of D-differentiability of interval-value mapping,and give the concept of corresponding gradient and related properties.As an application of D-differentiability,by discussing the optimality conditions of unconstrained interval-value programming problems whose objective mapping is D-differentiable interval-value mapping,we give the optimality conditions of a class of constrained interval-value programming problems whose objective mapping is D-differentiable interval-value mapping,and give the corresponding examples;At the same time,we discuss the problem of convexinterval-value programming with constraints as real-valued functions,and obtain sufficient conditions for optimal solutions.3.This paper discuss the sub-differentiability of interval-value mapping,give the concept and basic properties of sub-differentiability,and prove the sub-differentiability of interval-value mapping to be an empty set or a closed convex set.As an application of sub-differential,the relationship between the sub-differentiable of interval-value mapping and the sub-differentiable of convex interval-value mapping is discussed,and the sufficient conditions for the existence of convex extension of interval-value mapping under semi-continuous conditions are given.