CAJ | 학술논문

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.
본문대n유구씨공간중적겁소화문제전개연구,토론기Lagrange대우적부분기본성질,득출료관우Lagrange대우함수적량개신결과.수선증명재일반비공집합중,필정존재모일원소가용래표시해대우함수재임하일점적방향도수;연후,재차기출상득도료상비일개원상관경전정리경단순、경직접,집합소함원소위동류형차제도적결과,종이장Lagrange대우함수적방향도수표시진일보간화.
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 directional 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.