半(p,r)-(预)不变凸函数及其规划的鞍点最优性条件

Semi(p,r)-(Pre)Invexity Functions and Saddle Point Optimality Conditions for Programming with Semi(p,r)-Invexity

首先,定义了一类广义凸集——半p-不变凸集,在此基础之上,利用半预不变凸函数和(p,r)-预不变凸函数,定义了一类新的广义凸函数——半(p,r)-(预)不变凸函数,并举例说明了它既是半预不变凸函数又是(p,r)-预不变凸函数的真推广,从而是熟知的凸函数和不变凸函数的推广形式.接着,介绍了一个广义Lagrange向量函数L(x,u).最后,利用半(p,r)-不变凸函数讨论了多目标分式规划问题的鞍点最优性条件,得到了几个鞍点的存在性定理,其结论窟有一般性,推广了许多涉及不变凸函数,半预不变凸函数和(p,r)-(预)不变凸函数的文献的结论.

First, a class of generalized convex set called semi p- invex set is defined and based on preinvexity functions and （p, r） -preinvexity functions, a class of new generalized convex functions this, by using semi- called semi （p, r） - （pre） invexity functions are defined. It is showed with the aid of some examples that it is real generalization of the semi- preinvexity functions and （p, r ） -preinvexity functions, thus it is the generalization of well known convexity functions and invexity functions. Second, a vector-valued lagrangian L （ x, u） is introduced. At last, by using semi （p, r） -invexity functions, the saddle point optimality conditions of a muhiobjective fractional programming problem are studied, and some existence theorems for saddle point are derived. The work generalizes many results on programming problems with invex functions and semi-preinvexity functions, （p, r） -（pre）invex functions.