The Research For A Kind Of Generalized Convex Function With Its Optimization Problems

The properties of generalized convexity and directional derivative were used in this the-sis to discuss the relationships between the solutions of vector optimization problems and vector variational-like inequalities. And the relationships between invariant monotone maps and vector variational-like inequalities were discussed, we also introduced the concepts of perturbed generalized Stampacchia vector variational-like inequalities. Here was the sum-mary:In chapter one, we introduced the significance sense and situation of generalized con-vexity and variational inequalities.In chapter two, the notion of vector invariant pseudolinearity for non-differentiable and nonconvex functions was introduced by using the means of Dini directional derivatives. Some properties of vector pseudolinearity functions were discussed by using the concept of semi-order, including some characterizations of the solution set of a nonconvex and nondif-ferentiable, but invariant, pseudolinear program. The invariant pseudolinear and invariant weakly pseudolinear functions were introduced in this paper. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions and pseudolinear functions.In chapter three, some properties of pseudoinvex functions were given and relations between vector variational-like inequalities and vector optimization problems were discussed under pseudoinvexity and invariant pseudomonotonicity conditions, respectively.In chapter four, the relationships among the generalized Minty vector variational-like inequality problem(GMWLIP), generalized Stampacchia vector variational-like inequality problem(GSVVLIP) and vector optimization problems for nondifferentiable and nonconvex functions were discussed. We also considered the weak formulations of the generalized Minty vector variational-like inequality problems(WGMWLIP) and generalized Stampacchia vec-tor variational-like inequality problems(WGSVVLIP) and give some relationships between the solutions of these problems and a weak efficient solution of the vector optimization problems.In chapter five, we introduced the perturbed generalized weak Stampacchia vector variational-like inequality problems(PGWSWLIP).In chapter six, the main points of this paper were summarized.