库恩—塔克条件教学中两个容易忽略的问题

Two Easily Overlooked Problems in Teaching Kuhn-Tucker Conditions

库恩—塔克条件是同时含有等式与不等式约束函数的优化问题的最优性条件,已成为优化与控制、运筹与管理的一个核心内容。笔者通过多年的教学实践,发现了两个容易忽略的问题。第一个,对比经典的拉格朗日乘子法(针对于仅含等式约束函数的优化问题),它的推导方法并不是库恩—塔克条件的推导方法一个推广。第二个,库恩—塔克条件中的拉格朗日乘子(存在的话)未必是唯一的。

Kuhn-Tucker conditions are optimality conditions for optimization problems with equality and inequality constraint functions and are at the core of optimization and control,operations research and management.Through years of teaching,the authors found two easily overlooked problems.The first is related to understanding the relation between Kuhn-Tucker conditions and classical Lagrange method of multipliers for equality-constrained optimization problems:From their individual proof techniques,Kuhn-Tucker conditions can not be directly viewed as a generalization of the latter.The second is that Lagrange multipliers in Kuhn-Tucker conditions(if exist)is not necesssarily unique.