Characterizations Of ICR And DICR Functions And Applications Of Optimization Problems

In this dissertation,we consider a function φ:X × X× R++→[0,+∞]on a topological vector space.For every x,y ∈X,a∈R++,we definedφ(x,y,a)= sup{λ:0≤λ≤a,λX≤ y}.This function is a DICR function in the first argument and an ICR function in the second argument.We investigate these functions about several notions of abstract analysis such as subdifferential,support set and some related optimization problems.In this dissertation,the main work is as follows:In chapter 1,we introduce the research background,research motivations,situation at home and abroad,and main contents of this dissertation.We recalls some related concepts and conclusions which will be used in this dissertation.In chapter 2,we give the definition of ICR and DICR functions,present some characterizations for ICR and DICR functions,and also give relationships between these functions and abstract convexity.Furthermore,some relative results are also obtained about the functionφ(x,y,β)inf{λ:λ≥ β,λX≥y}.In chapter 3,we give the definition for subdifferential and support set of ICR and DICR functions.In addition,we investigate the relationship between subdifferential and support set of these functions,study the related properties.In chapter 4,we propose maximal elements of the set involving strictly DICR functions.As an application,we present the necessary and sufficient conditions for global minimum of the difference of two strictly DICR functions.