多目标最优化作为最优化领域的一个重要分支，着重研究在某些条件下多个数值目标的同时最优化的问题.多目标规划理论涉及凸分析、随机分析和泛函分析等一系列学科.在很多实际问题中，衡量一个方案的好坏往往难以用一个指标来判断，而需要用多个目标来比较，而这些目标有时是不协调，甚至是矛盾的.因此，对于多目标最优化问题的研究有着十分重要的现实意义.凸函数理论广泛应用于数学规划、控制论和复分析等领域.目前，不同广义凸性假设下的各类多目标规划的最优性和对偶性理论成果丰富，一些学者也持续对多目标优化进行深入研究.

本文主要定义一类新的不变凸函数，并在广义凸性假设下研究多目标规划以及多目标分式规划的最优性条件和对偶性结果.主要内容如下：

1.定义一类新的不变凸函数--H-(p,r)-Eta不变凸函数，在此广义凸性假设下，研究一类多目标规划问题的解的最优性，得到若干解的最优性充分条件；建立这类多目标规划的Wolfe型对偶以及Mond-Weir型对偶模型，证明相应的弱对偶、强对偶和严格逆对偶定理；

2.推广出H_b-(p,r)-Eta不变凸函数，在此广义凸性假设下，证明多目标分式规划解是弱有效解的最优性充分条件；建立多目标分式规划的Wolfe型对偶和Mond-Weir型对偶模型，证明相应的弱对偶、强对偶定理；

本文重新定义一类不变凸函数，得到了若干最优性条件和对偶性结论，丰富了现有的凸函数理论，充实了凸函数和多目标规划的相关成果，在理论上具有一定的研究意义.

Multi-objective optimization，as an important branch of optimization field，focuses on the simultaneous optimization of multiple numerical objectives under certain conditions.Multi-objective programming theory involves a series of disciplines such as convex analysis，stochastic analysis and functional analysis.In many practical problems, it is difficult to judge whether a scheme is good or bad with one index, but it needs to be compared with multiple objectives, which are sometimes uncoordinated or even contradictory. Therefore, the research on multi-objective optimization has very important practical significance. Convex function theory is widely used in mathematical programming，cybernetics and complex analysis.At present, various kinds of multi-objective programming under different generalized convexity assumptions have rich achievements in optimality and duality theory, and some scholars continue to conduct in-depth research on multi-objective optimization.

A new class of invexity functions is defined in this thesis，the optimality conditions and duality results of multi-objective programming and multi-objective fractional programming under the assumption of generalized convexity is researched.The main contents are as follows：

1.A new class of H-(p,r)-Eta invariant convex functions is defined.Under the assumption of generalized convexity，the optimality of solutions for a class of multi-objective programming problems is studied，and some sufficient conditions for optimality of solutions are obtained.Wolfe-type duality and Mond-Weir-type duality models of this kind of multi-objective programming are established，and the corresponding weak duality，strong duality and strict inverse duality theorems are proved.

2.The H_b-(p,r)-Eta invariant convex function is generalized，and under the assumption of generalized convexity，the optimality sufficient condition that the solution of multi-objective fractional programming is weakly efficient is proved.Wolfe-type duality and Mond-Weir-type duality models for multi-objective fractional programming are established，and the corresponding weak duality and strong duality theorems are prove.

In this thesis，a class of invariant convex functions is redefined，and some optimality conditions and duality conclusions are obtained，which enriches the existing convex function theory，enriches the related results of convex functions and multi-objective programming，and has certain research significance in theory.

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