Global Error Bounds And Metric Regularity For Almost Convex Inequality System In R~n

Error bound is an important research content in optimization theory.The global error bounds have significant applications in the sensitivity analysis of mathematical programming problems and the convergence analysis of various algorithms.Slater condition plays an major role in the characterization of global error bounds.Metric regularity has been regarded as one of the core concepts of contemporary variational analysis,which plays a very vital role in generalized equations,variational inequalities and optimization.In this dissertation,the global error bounds and relationship among metric regularity,global error bounds and Slater condition of almost convex inequality system in finite dimensional Euclidean spaces are proved by using the properties of almost convex sets and almost convex functions studied by Li and Mastroeni and employing the similar approach of Deng's error bounds result and the relationship among metric regularity,global error bounds and Slater condition.The dissertation consists of the following four chapters:In chapter 1,the research background and significance of error bounds and metric regularity for inequality system are introduced,as well as the structure of this dissertation.In chapter 2,some preliminary knowledge,some important properties and theorems are introduced.In chapter 3,we investigate global error bounds for almost convex inequality system in finite dimensional Euclidean spaces.By using the properties of almost convex sets and almost convex functions studied by Li and Mastroeni and employing the similar approach of Deng's error bounds result,we prove the global error bounds similar to Deng's results of almost convex inequality system in finite dimensional Euclidean spaces under certain conditions.We also give some examples to illustrate global error bounds for almost convex inequality system in finite dimensional Euclidean spaces.In chapter 4,we investigate the relationship among metric regularity,global error bounds and Slater condition for almost convex inequality system in finite dimensional Euclidean spaces.Employing the similar approach of Deng's results on metric regularity,global error bounds and Slater condition of convex inequality system in Banach spaces,the relationship among metric regularity,global error bounds and Slater condition of almost convex inequality system in finite dimensional Euclidean spaces is proved by theproperties of almost convex sets and almost convex functions obtained by Li and Mastroeni.We also prove some relationships between the global error bounds,metric regularity and Slater condition of convex inequality system and almost convex inequality system in finite dimensional Euclidean spaces.