In this paper, we present the notion of approximate solutions for vector variational inequalities in Banach spaces, which extends the existing approximate solutions. It is shown that, under the cone subconvexlikeness, our new approximate solutions can be characterized by the approximate solutions of linear scalar problem by means of the convex separation theorem. For the nonconvex cases, based on the nonconvex separation theorem with a special scalar functional introduced by Hiriart–Urruty, we get the nonlinear scalarization results for the approximate solutions. In order to establish optimality conditions without convexity assumption, we calculate the subdifferential of the Hiriart–Urruty nonlinear scalar functional. And based on the scalar characterizations, optimality conditions are obtained by using Ekeland variational principle and Fermat rule for scalar optimization problems. Finally, we consider the special case of vector inequality problems in finite dimensional Euclidean space, and establish some relations between the approximate solutions of vector inequality problems and nonsmooth vector optimization problems.

在本文中，我们提出了Banach空间中向量变分不等式的近似解的概念，从而扩展了现有的近似解。结果表明，在锥次凸似的情况下，利用凸分离定理，我们新的近似解可以用线性标量问题的近似解来表征。对于非凸情况，基于Hiriart–Urruty引入的具有特殊标量函数的非凸分离定理，我们得到了近似解的非线性标量结果。为了建立没有凸假设的最优条件，我们计算了Hiriart-Urruty非线性标量泛函的次微分。根据标量表征，利用Ekeland变分原理和Fermat规则求解标量优化问题，获得最优条件。最后，我们考虑了有限维欧几里德空间中向量不等式问题的特殊情况，并建立了向量不等式问题的近似解与非光滑向量优化问题之间的一些关系。