Convex Differential Analysis On The Infinite Dimensional Space And Perturbed Optimization And Variational Principle

The convex differential analysis in infinite-dimensional spaces has been studied almost seventy years. It has widely applications in many mathematics subjects, such as optimization theory and method, control theory, programing and global analysis. As we know, perturbed and optimization and varia-tional principle play an important role in nonlinear analysis. Mathematicians discussed various perturbed optimization and variational principle in Banach spaces or metric spaces since Ekeland's variational principle. But there are few general variational principle in more general topological linear spaces than Banach spaces. Through inducing a kind of nonlinear or sublinear topological space and the differentiability of convex function on it, we give the character-azition of perturbed optimization and variational principle of bounded sets in locally convex spaces. There are five chapters in this paper, our mainly results are as follows:As a way of resolving perturbed optimization and variational principle in locally convex spaces, we introduce a kind of sublinear topological space-Minkowski topological space and consider the differentiability of convex function in this space. Further, we make it to be a means of discussing perturbed optimization and variational principle in locally convex spaces and we haveSuppose that (X, p) is a Minkowski space, that C* is a nonempty w*-compact convex set with 0 C* C dp(0) for some p p, and suppose that C* has tu*-bata-exposed property. Then C* x R C (X x R, p}* also has the iy*-/3-exposed property.First we give the characterization of subsets on which every real-valued convex function bounded above is continuous on the space, and give the fi-perturbed optimization in Banach space by introducing new locally convex topology. Moreover, we give the characteristic theorem of function / having 6DP:Suppose that / is a continuous convex function on Banach space E andf* is the conjugate of / on E* (the dual of E). Then / has /J-differentiability property if and only if each level set of /* has w*-bata-exposed property.Applying the convexification of nonconvex function and the theorem in Chapter 2, we establish the perturbed optimization in the bounded subsets of the dual of locally convex spaces:Suppose that / is a real-valued w*-lower semicontinuous function defined on a unclosed bounded subset A* of the dual E* of a locally convex space E, which is bounded below on A*, and suppose every w* closed convex subset of C* = w*-dcoA* is the unclosed convex hull of its tu*-bata-exposed points. Then the bata-perturbed optimization is true in A*.We discuss the strong variational principle in locally convex spaces, present a geometric characterization of convex sets in locally convex spaces on which a strong optimization theorem of the Stegall-type holds, and give Collier's theorem about RNP a localized theorem.