凸度量空间中的广义凸性

Generalized Convexity in Convex Metric Spaces

针对凸度量空间中的抽象凸结构,提出了3种新广义W-凸函数,以及利用中点W-凸性研究了凸度量空间中广义凸性的方法。首先,将线性空间中基于标准凸结构的3种广义凸函数概念引入了凸度量空间,定义了3种新广义W-凸函数;其次,在适当条件下,证明了中间点W-凸函数是中点W-凸函数,也是[0,1]∩Q-W-凸函数,进而获得了稠密性定理,并讨论了稠密性定理在极小化问题和多目标规划问题中的应用;最后,在中点W-凸性以及上半连续性或下半连续性或W-拟凸性或W-严格拟凸性或W-半严格拟凸性等条件下,建立了W-凸函数的一些判别准则。获得的稠密性定理与利用中点凸性建立判别准则的方法,可以应用于其他类型凸性或广义凸性相关问题的研究。

For the abstract convex structure in convex metric spaces, three new generalized W-convex functions are proposed, and the method of using midpoint W-convexity to study generalized convexity in convex metric spaces is proposed. Firstly, the concepts of three generalized convex functions based on standard convex structure in linear space are introduced into convex metric space, and three new generalized W-convex functions are defined;Secondly, under some appropriate conditions, it is proved that the intermediate point W-convex function is a midpoint W-convex function, and the midpoint W-convex function is also [0,1]∩Q-W-convex function. Furthermore we obtain density theorem and discuss some applications of the density theorem in minimization and multiobjective programming problem;Finally, several discriminant criterions of W-convex function are established under the conditions of midpoint W-convexity and upper semicontinuity or lower semicontinuity or W-quasiconvexity or W-strict quasiconvexity or W-semi-strict quasiconvexity. The method of obtaining density theorem and establishing criterions by using midpoint convexity can be applied to the study of the related problems of other types of convexity or generalized convexity.