THEORY OF MAXIMA AND METHOD OF LAGRANGE
THEORY OF MAXIMA AND METHOD OF LAGRANGE
复制标题
DOI：
10.1137/0120037
复制
发表时间：
1971-01-01
影响因子：
1.9
通讯作者：
AFRIAT, SN
中科院分区：
文献类型：
--
作者：
AFRIAT, SN
1. 简介。 Lagrange (1762)(Courant [6, vol. I]) 用于确定具有约束的函数最大值的方法不区分最大值和最小值，也不区分最大值和绝对最大值。汉考克[11]最近提出了允许进行这种区分的进一步条件，但这种处理缺乏完整性。这个主题在经济学中具有重要意义，而萨缪尔森 (Samuelson) 的论述[17] 与汉考克 (Hancock) 的思路相同，最为人所熟知。 Frisch [10] 给出了进一步的阐述。这里对一些主要结果进行了处理。该方法通过两种途径涉及隐式函数定理的不同用途（Carath6odory [5]、Rudin [16]）。给出了该定理的证明，并推导出了适用于函数依赖和拉格朗日理论的推论。约束变量的最大函数的可微性在定义该函数的任何邻域中建立，并且其导数用拉格朗日变量来标识。更一般地，当目标函数和约束函数携带其他变量作为参数时，找到最大点和拉格朗日乘子相对于这些参数的导数的表达式（如果存在），并显示存在的条件。然后就参数而言，在最大值和拉格朗日函数的导数之间建立恒等式。这被视为概括了拉格朗日乘子与导数的通常识别。汉考克条件要求二次形式在线性约束下是确定的。他获得了这种条件的行列式形式，然后 Mann [15] 导出了最著名的更简单的行列式形式的条件。 Ferrar 8 处理了一种约束的特殊情况]。它在斯卢茨基[18]（希克斯[12，附录]）的消费者理论中得到了独特的进一步发展。 Afriat [1] 给出了另一个推导以及格拉斯曼坐标方面的对偶版本和相关表达式。这里提供了另一个推导。 Finsler [9] 获得了不同的条件形式，Afriat 2] 和 Bellman 4] 也考虑了这一点。在本文中，向量的 x> _ y 意味着每个元素 xi>= yi，x> y 意味着 x> y 但 x-y，x> y 意味着 x> y。又，R表示m阶列向量，即m 1 矩阵，R中的元素，R为实数； R 表示 n 阶实数行向量，即 1 n 矩阵，R 表示实数 m n 矩阵。
1. Introduction. The method of Lagrange (1762)(Courant [6, vol. I]) for determining a maximum of a function with constraints does not distinguish between a maximum and a minimum, nor between a maximum and an absolute maximum. Further conditions which allow such distinctions to be made have been brought into view more recently by Hancock [11], buthis treatment lacks completeness. The subject has importance in economics and the account of Samuelson 17], which is along the same lines as thatof Hancock, is most familiar. A further exposition is given by Frisch [10]. Here there is a treatment of some principal results. The approach is by two routes involving differentuses of the implicit function theorem (Carath6odory [5], Rudin [16]). A proof is given of that theorem, and corollaries are deduced which have application to functional dependence and Lagrangian theory. The differentiability of the maximum functionof the constraint variables is established in any neighborhood where the function is defined, and its derivatives are identified with Lagrangian variables.More generally, when the objective and constraint functions carry further variables as parameters, expressions are found for the derivatives of the maximum point and of the Lagrangian multipliers with respect to these parameters, when these exist, and conditions for existence are shown. Then the identity is established between the derivatives of the maximum value and of the Lagrangian function, with respect to the parameters. This is seen to generalize the usual identification of Lagrangian multipliers with derivatives. Hancock’s conditions require a quadratic form to be definite subject to linear constraints. He obtained a determinantal form for such a condition, and then Mann [15] derived the condition in thesimpler determinantal form which is most well known. The special case of one constraint is dealt with by Ferrar 8]. It has a peculiar further development in the consumer theory of Slutsky [18](Hicks [12, Appendix]). Another derivation together with a dual version and related expressions in terms of Grassmann coordinates is given by Afriat [1]. Here another derivation is provided. Finsler [9] obtained a different form for the conditions, considered also by Afriat 2] and Bellman 4]. Throughout this paper, x> _ y for vectors means xi>= yi for each element, x> y means x> y but x-y, and x> y means x> y. Also, R denotes the column vectors of order m, or the m 1 matrices, with elements in R, R being the real numbers; R, denotes the real row vectors of order n, or 1 n matrices, and R denotes the real m n matrices.