Error Bound And Perturbed Analysis For Infinite Inequality Systems In Banach Space

In this paper, we consider error bound and perturbed analysis for infinite inequality systems in Banach space, including the following two problems:1、Local error bound for infinite convex-composite inequality systems. Let I be an arbitrary index set and X,Xi(i ∈ I) are Banach spaces. Consider the infinite convex-composite inequality systemwhere for each i ∈ I, Fi : X→ Xi is a Frechet differentiable map and hi : Xi→ R :=R ∪{+∞}is a proper lower semicontinuous convex function.2.Error bound for abstract conic inequality systems. Let X, Y be Banach spaces.Consider the abstract conic inequality systems.where F : →Y is a smooth map, K (?) Y is a nonempty convex cone(not necessarily close set).We always assume X, Y are Banach spaces, we establish the subdifferential for func-tion by using method of nonsmooth analysis and variational analysis, and in terms of the normal cones of the solution set and the subdifferential of the concerned function in the so-lution set, we derive the sufficient condition and necessary condition of local error bound for the infinite convex-composite inequality systems at some point. As application, we es-tablish the characterization of linear regularity for infinite set family.Under the weak assumption, we study error bound for abstract conic inequality sys-tems and error bound for perturbed abstract conic inequality systems by using Simple New-ton method.The background and the main works are organized as follows.In Chapter 2, We list some notations, some notions and some basic results about non-smooth analysis, variational analysis and optimization theory.In Chapter 3, We study the infinite convex-composite inequality system. We introduce the definition of strictly differentiable and lp type, q1 type uniformly strictly differentiable and p1 type uniformly Lipschitz. By using nonsmooth analysis and variational analysis,we establish the subdifferential. Granting this, we obtain the sufficient condition and nec-essary condition of local error bound for infinite convex-composite inequality system. As application, we study the linear regularity for infinite set family and the equivalent charac-terization. Most of our results are new and extend the corresponding conclusions[97,99].In Chapter 4, we study abstract conic inequality systems. By using conceptions of convex process, weak-Robinson conditions and center-Lipschitz continuous etc, we prove convergence results of the simple Newton method for solving the inequality system. Based on the convergence results, we establish the error bound for abstract conic inequality sys-tems and error bound for perturbed abstract conic inequality systems. As application, we consider the perturbed linear inequality systems and obtain some results. Most of our re-sults are new and extend the corresponding conclusions[57,59,61].