Well-posedness Of Split Minimization Problems

The issue of well-posedness is one of the important subjects in the theory of various variational problems.The well-posedness is essentially the study of the convergence of various approximation solution sequences,and it has important applications in the convergence analysis of algorithms.In this thesis,we study the well-posedness of split minimization problems.In Chapter 2,we investigate the Levitin-Polyak well-posedness by perturbations of split minimization problems.We extend well-posedness notions to the split minimization problem and prove that the split minimization problem in the setting of finite-dimensional spaces is Levitin-Polyak well-posed by perturbations provided that its solution set is nonempty and bounded.We also extend well-posedness notions to the split inclusion problem.We discuss the relations between the well-posedness of the split minimization problem and the well-posedness of the split inclusion problem.In Chapter 3,we study the generalized Levitin-Polyak well-posedness of split constrained minimization problems.We introduce a split constrained minimization problem and extend Levitin-Polyak well-posedness,generalized Levitin-Polyak well-posedness and strongly generalized Levitin-Polyak well-posedness to the split constrained minimization problem.Under the appropriate conditions,some necessary and/or sufficient conditions for three types of well-posedness are given.We derive metric characterizations of Levitin-Polyak well-posedness and generalized Levitin-Polyak well-posedness.Finally,we investigate the relations among three types of well-posedness.