Cone Constrained Convex Vector Optimization Problems With Polyhedral Control Cone Efficient Solution Set Nonempty Boundary Of The Characterizations

In real world, the importance of (vector) optimization problems has been widely recognized. More and more people who study mathematics and engineering are interested in it. Its theories and methods are widely used in the areas of mathematical programming, management science, engi-neering, the system of social economic and etc. Especially in view of the close relationship between vector optimization problems and the decision-makers' preferences, the study of the characteriza-tions of (weakly, properly)efficient solutions has great value in theory and practical significance.The main research object of this paper is the characterizations of the efficient solution set of a class of cone-constrained convex vector optimization problem with a domination cone. In this paper, we generalize the existing conclusions of a vector optimization problem with the Pareto domination cone, characterize the nonemptiness and boundedness of the efficient solution set of a cone-constrained convex vector optimization problem whose domination cone is polyhedral with nonempty interior, and then, we apply a key condition in the characterizations to the convergence analysis of a class of penalty methods.The main research works are as follows. In the first part of this paper, we introduce the background and the development process of the vector optimization problem, and give the the pre-liminary knowledge, including the basic concepts and some relevant conclusions. Then in charpter one, we characterize the nonemptiness and compactness of optimal solution set of a single objective optimization problems. In charpter two, we introduce existing research results of nonemptiness and compactness of the weakly efficient solution set of a vector optimization problem in a finite dimen-sional space and their applications in the convergence of a class of penalty methods. In charpter three, we generalize the previous conclusions of a vector optimization problem with the Pareto domination cone, characterize the nonemptiness and boundedness of the efficient solution set of a cone-constrained convex vector optimization problem whose domination cone is polyhedral with nonempty interior, and then, we apply a key condition in the characterizations to the convergence analysis of a class of penalty methods. In charpter four, major achievable outcomes and significance of this research are summarized.