In this paper, we study a generalized convex vector equilibrium problem with cone and set constraints in real Banach spaces. We provide some basic characterizations on generalized convexity for the first- and second-order directional derivatives. We obtain Kuhn–Tucker second-order necessary and sufficient optimality conditions for efficiency to such problem under suitable assumptions on the generalized convexity of objective and constraint functions. As an application, we present Kuhn–Tucker second-order necessary and sufficient optimality conditions to a generalized convex vector variational inequality problem and a generalized convex vector optimization problem with constraints. Some examples are also given to demonstrate the main results of the paper.

在本文中，我们研究了在实际Banach空间中具有锥和集约束的广义凸矢量平衡问题。我们提供了有关一阶和二阶方向导数的广义凸的一些基本特征。我们在目标和约束函数的广义凸性的适当假设下，获得了针对此类问题的效率的Kuhn-Tucker二阶必要和充分的最优条件。作为应用，我们为广义凸向量变分不等式问题和带约束的广义凸向量优化问题提供了Kuhn-Tucker二阶必要和充分的最优性条件。还给出了一些例子来证明本文的主要结果。