LAGRANGE MULTIPLIERS AND OPTIMALITY
LAGRANGE MULTIPLIERS AND OPTIMALITY
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DOI：
10.1137/1035044
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发表时间：
1993-06-01
期刊：
影响因子：
10.2
通讯作者：
ROCKAFELLAR, RT
中科院分区：
文献类型：
--
作者：
ROCKAFELLAR, RT
拉格朗日乘子被视为辅助变量引入到约束最小化问题中，以便将一阶最优性条件正式地写成方程组。现代应用，其重点是数值方法和更复杂的侧边条件比方程，要求更深入的理解的概念，以及如何将其融入一个更大的理论图片。一个主要的研究路线一直是非光滑几何的单边切线和法向量的点集满足给定的约束。另一个是乘数向量作为对偶问题的解的博弈论作用。解释为广义衍生物的最优值相对于问题参数也进行了探讨。拉格朗日乘数现在被看作是产生于一个非光滑的目标函数，允许黑和白色的限制，以取代罚款表达式的次微分的一般规则。本文跟踪这些主题在当前的理论拉格朗日乘数，提供沿着的方式一个独立的阐述基本非光滑分析的动机和应用于这一主题。
Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture.A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a free-standing exposition of basic nonsmooth analysis as motivated by and applied to this subject.