最优化问题的二阶最优性条件中临界锥不同表达形 式的研究

在不同的最优化方法教材中，优化问题的二阶最优性条件中的临界锥有多种不同的表 达形式，使得初学者在接触这部分知识时可能会感到困惑，实际上这些表达形式在一定条件下是 等价的。文章总结了不同教材中临界锥的六种表达形式，论证了它们之间的联系，给出了这些临 界锥相等的充分条件，使得不同形式的二阶最优性条件有了统一的表达形式。

Study on Different Expressions of Critical Cone in Second Order Optimality Conditions of Optimization Problems

In different textbooks about optimization methods, there are many different expressions of the critical cone in the second-order optimality conditions of optimization problems. Beginners may be confused when contacting this part of knowledge. In fact, these expressions are equivalent under certain conditions. In this paper, six expressions of critical cones in different textbooks were summarized, and the relations between them were demonstrated. The sufficient conditions for the equality of these critical cones were given, so that different forms of second-order optimality conditions have a unified expression.