论文标题
重新访问平等控制最小化问题的二阶最佳条件
Revisiting second-order optimality conditions for equality-contrained minimization problem
论文作者
论文摘要
本注释的目的是对等值限制的最小化问题的经典二阶最佳条件进行几何见解。我们表明，在当地的最低点$ x^*$相关的Hessian对与问题相关的拉格朗日功能的积极性对应于hypersurface $ \ Mathcal $ \ Mathcal {m} _} _} _ {m} _ {f，x^*} =^*} = f，x^*} = f，x^*}的相应代数曲率之间的不平等现象\，f（x）= f（x^*）\} $由目标函数$ f $定义和submanifold $ \ mathcal {m} _g = \ = \ {x \ in \ r^n \，| \，g（x）= 0 \} $定义相反。这些不等式突出了几何证据，以确保最佳性，submanifold $ \ mathcal {m} _g $必须本地包含在半个空间$ \ mathcal {m} _ {m} _ {f，x^*}^*}^*}^*}^+ = \ f， \，f（x）\ geq f（x^*）\} $受hypersurface $ \ mathcal {m} _ {f，x^*}的限制。$此演示文稿可用于教育目的，并有助于更好地理解此属性。
The aim of this note is to give a geometric insight into the classical second order optimality conditions for equality-constrained minimization problem. We show that the Hessian's positivity of the Lagrangian function associated to the problem at a local minimum point $x^*$ corresponds to inequalities between the respective algebraic curvatures at point $x^*$ of the hypersurface $\mathcal{M}_{f, x^*}=\{ x \in \R^n \, | \, f(x) = f(x^*)\}$ defined by the objective function $f$ and the submanifold $\mathcal{M}_g = \{ x \in \R^n \, | \, g(x)= 0 \}$ defining the contraints. These inequalities highlight a geometric evidence on how, in order to guarantee the optimality, the submanifold $\mathcal{M}_g$ has to be locally included in the half space $\mathcal{M}_{f, x^*}^+ = \{ x \in \R^n \, | \, f(x) \geq f(x^*)\}$ limited by the hypersurface $\mathcal{M}_{f, x^*}.$ This presentation can be used for educational purposes and help to a better understanding of this property.
