Convexity Of Sets And Its Applications

In 1960s there formed a new mathematical branch subject---convex analysis, which focuses on convex sets and convex functions. Now convexity theory becomes an important theoretic fundamental and a useful tool of mathematical programming, variation calculus, and optimization theory. For the need of solving practical problems, people have generalized the convexity from different point of view. However, convex analysis always plays a necessary role in the mathematical development. Therefore, studying convexity and its applications in optimization is still important and interesting. Because of the above, properties of functional operations will be discussed in a new way according to convexity and operational properties of sets. Moreover the epigraph will be discussed in order to characterize functions intuitively.Chapter 1 acts as the general introduction to the significance and current situation in the study of convex analysis.Chapter 2 discusses some new properties about sets especially convex sets, based on some theorems concerning convex analysis. Next, sets' operations such as convex hull, interior and relative interior operations are discussed. Some results in R~n are generalized to a linear topological space; new conditions assuring int A + B = int(A+B) are given, and conclusions about interior are generalized to the situation of relative interior. At last, a few conditions are given to assure n(A + B)(?)riA + B.Chapter 3 specially studies the epigraph and its properties in order to apply characters of sets to functions, and obtains a conclusion that a function can be characterized by its epigraph i.e.f(x) = inf{μ| (x,μ)∈epif}, then defines some new functional operations like interior and relative interior operations of functions. Next, functional operations such as convex hull, right scalar multiplication, interior and relative interior operations are discussed. At last, properties about sets are applied to functions with respect to the intimate link between the function and its epigraph.Chapter 4 comes to a conclusion. In the meanwhile, it puts forward some problems for further study.In this thesis, main results gather in chapter 2 and 3, and new results gather in the 5-th, 6-th and 7-th of chapter 2 and in all of chapter 3.