Abstract:
In this paper, various characterizations of optimal solution sets of nonsmooth B-preinvex optimization problems with inequality constrains are given. Firstly, making use of Clarke’s subdifferential, we establish the optimality condition for this kind of optimization problem; secondly, we presented a property about the solution set S of constrained B-preinvex optimization proble; finally, five equivalent characterizations of the solution set are obtained, that is,* An example is given to illustrate that five solution sets are equal, i.e. S=｛0｝.(* Indicates a formula, please see the full text)

Abstract:
We introduce some new classes of preinvex and invex functions, which are called ϕ-preinvex and ϕ-invex functions. We study some properties of these classes of ϕ-preinvex (ϕ-invex) functions. In particular, we establish the equivalence among the ϕ-preinvex functions, ϕ-invex functions, and ϕη-monotonicity of their differential under some suitable conditions.

Abstract:
We introduce some new classes of preinvex and invex functions, which are called -preinvex and -invex functions. We study some properties of these classes of -preinvex ( -invex) functions. In particular, we establish the equivalence among the -preinvex functions, -invex functions, and η -monotonicity of their differential under some suitable conditions.

Abstract:
The paper gives a class fo new generalized convex function-strongly G-preinvex functions, it is a true generalization of strong preinvex function. First, three, examples have been got to show that it's existence, and strongly G-preinvex function is different from G-preinvex function and strictly G-preinvex function. Then, we discuses three properties of strongly G-preinvex function. Finally, we give a sufficient condition about strongly G-preinvex function under the case that G-preinvex function,namely, Let the set * is invex set with * is satisfied with condition * is a G-preinvex function. if * have * . Then, * is a strongly G-preinvex function on K with respect to *.(* Indicates a formula, please see the full text)

Abstract:
The present paper deals with the properties of geodesic -preinvex functions and their relationships with -invex functions and strictly geodesic -preinvex functions. The geodesic -pre-pseudo-invex and geodesic -pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.

Abstract:
A new type of generalized convex functions, termed preinvex functions, is further discussed in this paper. A new result that if a preinvex function satisfies intermediate-point semistrict preinvexity then it is also a semistrictly preinvex function is obtained. Furthermore, another simplified proof for a criterion of preinvex functions under weaker conditions and an impotant application of preinvex functions are given.

Abstract:
本文研究了非凸集值向量优化的严有效解在两种对偶模型的强对偶问题.利用Lagrange对偶和Mond-Weir对偶原理，获得了如下结果：原集值优化问题的严有效解，在一些条件下是对偶问题的强有效解，并且原问题和对偶问题的目标函数值相等；推广了集值优化对偶理论在锥-凸假设下的相应结果. This paper is diverted to the study of two strong dual problems of a primal nonconvex set-valued optimization in the sense of strict efficiency.By using the principles of Lagrange duality and Mond-Weir duality,for each dual problem,a strong duality theorem with strict efficiency is established.The conclusions can be formulated as follows:starting from a strictly efficient solution of the primal problem,it can be constructed a strictly efficient solution of the dual problem such that the corresponding objective values of both problems are equal.The results generalize the strong dual theorems in which the set-valued maps are assumed to be cone-convex

Abstract:
We firstly construct a concrete semi-invex set which is not invex. Basing on concept of semi-invex set, we introduce some kinds of generalized convex functions, which include semi-(,)-preinvex functions, strictly semi-(,)-preinvex functions and explicitly semi-(,)-preinvex functions. Moreover, we establish relationships between our new generalized convexity and generalized convexity introduced in the literature. With these relationships and the well-known results pertaining to common generalized convexity, we obtain results for our new generalized convexities. We extend the existing results in the literature.

Abstract:
Generalized convexity has been playing an important role in mathematical programming . In this paper, an equivalent condition of twice continuously differentiable preinvex function is established by transforming multivariate real-valued function into univariate real-valued function. Suppose that X be open invex set with respect to η,ηsatisfies condition C , f be twice continuously differentiable and satisfies condition D. Then f is preinvex function with respect to η,ηsatisfies condition C, f be twice continuously differentible and satisfiles condition D. then f is preinvex function with respect to η if and only if νx,y∈X,η(x,y)tV2f(x)η(x,y)≥0. Our results provide new thoughts to verify the preinvexity of function and also generalize some known results.