多目标规划是应用数学和决策科学的一个交叉学科，凸函数是金融学、数理统计学和最优化理论的基础。在多目标规划问题中，大部分的结果都受目标函数和约束函数的凸性限制，但是由于凸函数具有一定的局限性，而在我们所遇到的实际问题中大量的函数是非凸的，因此对凸函数的推广即广义凸函数是众多学者研究的热点课题。本文通过引入不变凸函数来进一步讨论多目标规划中的有关问题，不变凸性在一定程度上既保留了凸函数的优良性质，同时也是凸函数的拓广和发展。

在前人工作的基础上，本文对凸函数作了多种形式的推广，提出了一类新的广义高阶不变凸性概念，并研究了目标函数和约束条件都是新广义高阶不变凸函数的多目标规划和多目标分式规划的最优性条件、对偶性结果和鞍点问题。主要内容如下：

(1)首先定义了一类新的广义高阶(F,η)-不变凸函数，并通过恰当的例子验证其正确性。其次，在新广义凸性假设条件下，研究了多目标分式规划的最优性，得到了一些最优性充分条件和鞍点理论。

(2)构造了高阶(F,η)-不变凸多目标分式规划对应的Mond-Weir型和Wolfe型对偶模型，分别得到并证明了相应的弱对偶、强对偶和逆对偶定理。

(3)进一步构造了更接近最优解的多目标规划的高阶Mond-Weir型和高阶Wolfe型对称对偶模型，在广义高阶(F,η)-不变凸性假设下，分别得到并证明了若干相应的对偶结果。

Multi-objective programming is an interdisciplinary subject of applied mathematics and decision science. Convex functions are the basis of finance, mathematical statistics, and optimization theory. In multi-objective programming problems, most of the results are limited by the convexity of the objective function and the constraint function. However, because convex functions have certain limitations, and a large number of functions are non-convex in the practical problems we faced, therefore, the generalization of the convex function, namely the generalized convex function, is a hot subject studied by many scholars.This paper introduces invex functions to further discuss related problems in multi-objective programming. The generalized convexity not only retains the excellent properties of convex functions, but also extends and develops convex functions.

 On the basis of previous work, this paper makes a variety of generalizations of convex functions, this paper presents a new class of generalized higher-order convexity concepts, and studies the conditions for optimality, duality conclusions, and saddle-point problems of multi-objective programming and multi-objective fractional programming in which both the objective function and the constraints are new generalized higher-order convex functions.The main contents are as follows:

(1) Firstly, a new class of generalized higher-order (F,η)-invex functions is defined, and its correctness is verified through appropriate examples. Secondly, under the condition of the new general convexity assumption, the optimality of the multi-objective fractional programming is studied, and some optimality sufficient conditions and saddle point theory are obtained.

(2) The Mond-Weir type and Wolfe type dual models corresponding to higher-order (F,η)-invex multi-objective fractional programming are constructed, and the corresponding weak dual, strong dual, and inverse dual theorems are obtained and proved respectively.

(3) Under the generalized higher-order (F,η)-invexity assumption, the higher-order Mond-Weir type and higher-order Wolfe type symmetric dual models of multi-objective programming are further constructed, several corresponding dual results are obtained and proved.