半定锥约束变分不等式的二阶充分性条件

Second-Order Sufficient Conditions for Semi-Definite Constrained Variational Inequalities

摘要: 半定锥约束变分不等式问题是目前研究热门话题——锥约束变分不等式问题的其中一部分，在优化理论、经济学和工程应用中具有重要意义。在本文中，我们给出了半定锥约束变分不等式问题的二阶充分性条件。我们首先回顾了变分不等式的基本理论概念及其在半定锥约束下的具体形式，然后通过构造相应的拉格朗日函数，将问题的解与二阶充分性条件关联起来，最后我们给出了相关定理的具体内容及其证明，确保在给定约束条件下解的存在性与最优性。

Abstract: The semi-definite cone-constrained variational inequality problem is currently a hot topic of research and is a part of the broader cone-constrained variational inequality issues, which hold significant importance in optimization theory, economics, and engineering applications. In this paper, we present the second-order sufficient conditions for the semi-definite cone-constrained variational inequality problem. We first review the fundamental theoretical concepts of variational inequalities and their specific forms under semi-definite cone constraints. Then, by constructing the corresponding Lagrangian function, we establish a connection between the solutions of the problem and the second-order sufficient conditions. Finally, we provide the specific content and proof of the relevant theorems to ensure the existence and optimality of the solutions under the given constraints.