预不变凸、半严格预不变凸函数的性质与应用

Properties of preinvex and semistrictly preinvex functions and their application

研究两类重要的广义凸函数—预不变凸函数、半严格预不变凸函数.得到关于预不变凸、半严格预不变凸函数的两个重要的性质定理:1)在一定条件下,f是预不变凸函数的充分条件为f是半严格预不变凸函数且满足弱中间点预不变凸性;2)在一定条件下,f是严格预不变凸函数的充分条件为f是半严格预不变凸函数且满足弱中间点严格预不变凸性,此结果将凸函数的相应结果推广到预不变凸函数情形.给出预不变凸函数在极小化问题中的一个重要应用.

Two important types of generalized convex functions termed as preinvex functions and semistrictly preinvex functions were studied. Two important theorems of properties were obtained for these. Firstly, two important conclusions of preinvex functions were obtained, namely 1） under a certain condition, a semistrictly preinvex function satisfies intermediate-point weak preinvexity and was the sufficient condition for a preinvex function; 2） under a certain condition, a semistrictly preinvex function satisfies in- termediate-point weakly strict preinvexity and was the sufficient condition for a strictly preinvex function. This result generalized the theorem for convex functions to the preinvex ones, presenting an important application of preinvex functions in the problem of minimization.