With Cone Constrained Optimization Problem Of Weak Sharp

The notion of sharp minima, or strongly unique local minima due to B. T. Polyak [88], emerged in the late 1970s as an important tool in the analysis of the perturbation behavior of certain glasses of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. The related study of sharp minima for single objective optimization problems of L. Cromme and B. T. Polyak is of particular importance in this development. In the late 1980's M. C. Ferris coined the term weak sharp minima to describe the extension of the notion of sharp minima to include the possibility of a non-unique solution set. The study of weak sharp minima is motivated primarily by applications in convex, and convex composite programming, where such minima commonly occur. For example, such minima frequently occur in linear programming, linear complementarity, and least distance or projection problems. In this paper, we study generalized weak sharp minima for convex optimization problems with cone constraints and weak (?)-sharp minima for multi-objective optimization problems with cone constraints. The main research works are as follows.In chapter two, we introduce a notion of generalized weak sharp minima for convex optimization problems with cone constraints. We study it in the Banach space and Hilbert space settings, respectively. Several characterizations for the solution set to be a set of generalized weak sharp minima for convex optimization problems with cone-constraints are provided. In particular, under the assumption that Robinson's constraint qualification holds over the solution set, we give several necessary and sufficient conditions for the generalized weak sharp minima property of convex optimization problems with cone constraints. As applications, we propose an algorithm for convex optimization problems with cone constraints in the Hilbert space setting. Convergence analysis of this algorithm is given.In chapter three, We first introduce a notion of local weak sharp solutions for convex-composite optimization problems given by Zheng and Ng [105]. We find that this definition is very limited in applying to the usual mathematical program-ming. Even for linear programming, it is shown that the solution set is not that type of local weak sharp solutions. In this section, we consider a class of convex optimization problem with cone constraints in Banach spaces. We introduce a new notion:type I generalized (local) weak sharp minima for the convex conic optimiza-tion problem. Several types of strong Karush-Kuhn-Tucker characterizations and criteria for a feasible point to be a type I generalized local weak sharp minimum are given. As applications, local error bounds for a class of non-degenerate conic differential convex inclusion problem are studied.In chapter four, we study generalized well-posedness for multi-objective opti-mization problems with cone constraints in the case when the decision space X is a finite dimensional normed space and the object space Y is a normed vector space ordered by a closed convex cone with nonempty interior or a polyhedral cone with nonempty interior. Our results generalize most of the corresponding ones established by Deng [26]. Finally, under mild conditions, we also show that general linear vector optimization problems, a class of piecewise linear vector optimization problems and a class of convex vector quadratic programs are generalized well-posed.In chapter five, we study weak (?)-sharp minima for multi-objective optimiza-tion problems with cone constraints in the Banach space. Several sufficient or necessary conditions for existence of such class of weak sharp minima are obtained. The relationships between weak (?)-sharp minima and a type of well posedness for multi-objective optimization problems with cone constraints are established. As applications, we prove upper Holder continuities of the solution set-valued mapping to the corresponding parametric multi-objective optimization problems with cone constraints.