Vector OptinUzation Of The Cone And Minty Variational Inequalities

Vector optimization problem which has nonsmooth objective function, is a problem that requires multiple objectives to achieve optimal at the same time under some given conditions. Under convex and differentiable conditions, numerical optimization problem is equal to a classical variational inequalities. Since Minty proposed the concept of the Minty, Minty variational inequality theories have been extended continuously. In this dissertation, the Minty variational inequalities which is closely related to the vector optimization problems is discussed, the relationship of Minty variational inequalities and vector optimization problems is established. By defining new derivative and convexity, vector optimization problems of compactness, smoothness and convexity is weakened, finite-dimensional space is extended to the infinite-dimensional normed space, then vector optimization problems of the cone are discussed, these results extend some corresponding results in the litcrature. The main results of this dissertation consist of the following five parts:In chapter1, we recall the academic background and the development of vector optimization problems and variational inequalities, then introduce the main content and the structure of the investigation.In chapter2, by defining of the directional derivative, the relationship of Minty vector variational inequality (Minty ⅤⅥ), Stampacchia vector variational inequality (Stampacchia ⅤⅥ) and vector optimization problem (VOP) is discussed.In chapter3, stability of a parametric quasivariational cone of the Minty type is studied via various sufficient conditions characterizing upper and lower semicontinuity of the solution sets as well as the approximate solution sets, which is similar to the solution sets of inequality. Sufficient conditions ensuring upper semicontinuity of the approximate solution sets of an optimization problem with quasivariational cone constraints are also presented.In chapter4, under nondifferentiable and nonconvex mapping, we study the relationship among the generalized Minty vector variational-like inequality, generalized Stampacchia vector variational-like inequality and vector optimization problem. The weak formulations of the above has also been considered and some relationships between the solutions of these and a weak efficient solution of the vector optimization problem has been given.In chapter5, the relationship of generalized type-I, generalized quasi type-I, generalized pseudo type-I functions over cones are defined by differential theory. Sufficient optimality conditions and the nature of the dual are also studied.