In this paper, we study the Fritz John necessary and sufficient optimality conditions for weak efficient solutions of vector equilibrium problem with constraints via contingent hypoderivatives in finite-dimensional spaces. Using the stability of objective functions at a given optimal point and assumming, in addition, that the regularity condition (RC) holds, some primal and dual necessary optimality conditions for weak efficient solutions are derived. Furthermore, a dual necessary optimality condition is also established for the case of Fréchet differentiable functions. Making use of the concept of a support function on the feasible set of vector equilibrium problems with constraints, some primal and dual sufficient optimality conditions are given for the class of stable functions and Fréchet differentiable functions at a given feasible point. As an application, several necessary and sufficient optimality conditions for weak efficient solution are also obtained with the class of Hadamard differentiable functions. Examples to illustrate our results are provided as well.

中文翻译：

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使用或然次导数约束向量平衡问题的充要条件

在本文中，我们通过有限维空间中的或有次导数研究了带约束的矢量平衡问题的弱有效解的Fritz John充要条件。利用目标函数在给定最佳点的稳定性，并假设正则条件（RC）成立，得出了弱有效解的一些原始和对偶必要最优条件。此外，对于弗雷谢特微分函数的情况，还建立了双重必要的最优性条件。利用带有约束条件的向量平衡问题的可行集上的支持函数的概念，给出了给定可行点上稳定函数和Fréchet可微函数类的一些原始和对偶最优性条件。作为应用，还利用Hadamard可微函数类获得了弱有效解的几个必要条件和充分最优条件。还提供了一些例子来说明我们的结果。