Lagrange函数方向导数的简化表示

A Simplified Expression for Directional Derivative of Lagrangian Function

本文对n维欧氏空间中的极小化问题展开研究,讨论其Lagrange对偶的部分基本性质,得出了关于Lagrange对偶函数的两个新结果.首先证明在一般非空集合中,必定存在某一元素可用来表示该对偶函数在任何一点的方向导数;然后,在此基础上得到了相比一个原相关经典定理更单纯、更直接,集合所含元素为同类型次梯度的结果,从而将Lagrange对偶函数的方向导数表示进一步简化.

This paper investigates the minimization problem in n^dimensional Euclidean space, and analyzes some basic properties of the Lagrangian dual of the problem. Two new results on the Lagrangian dual function are further obtained. We first prove that there exists the element in an arbitrary nonempty set which can be utilized to formulate the directionM derivative of the Lagrangian dual function at any given point. And then we obtain a simpler and more straight- forward conclusion that there only exists the same type of subgradients compared with that in a classic result. The mathematical expression of the directional derivative of the Lagrangian function is thus simplified.