Scalarization and Optimality Conditions for the Approximate Solutions to Vector Variational Inequalities in Banach Spaces
Scalarization and Optimality Conditions for the Approximate Solutions to Vector Variational Inequalities in Banach Spaces
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DOI：
10.1007/s41980-020-00507-1
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发表时间：
2021-03
影响因子：
0.7
通讯作者：
Ying Gao;Rui-Xue Yue;L. Tang
中科院分区：
文献类型：
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作者：
Ying Gao;Rui-Xue Yue;L. Tang
本文提出了Banach空间中向量变分不等式的近似解的概念，推广了已有的近似解。证明了在锥次类凸条件下，利用凸分离定理，我们的新近似解可以用线性标量问题的近似解来刻画。对于非凸情形，利用Hiriart-Urruty引入的具有特殊标量泛函的非凸分离定理，得到了近似解的非线性标量化结果.为了建立不需要凸性假设的最优性条件，我们计算了Hiriart-Urruty非线性标量泛函的次微分。在此基础上，利用Ekeland变分原理和Fermat规则，得到了标量优化问题的最优性条件。最后，我们考虑了有限维欧氏空间中向量不等式问题的特殊情况，建立了向量不等式问题的近似解与非光滑向量优化问题的近似解之间的关系。
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.