多元函数条件极值的一种较精确的充分条件

On a More Precise Sufficient Condition for Conditional Extreme Values of Multi-variable Functions

无条件极值的充分条件采用其目标函数的Hesse矩阵作为判别依据,但多元函数条件极值充分条件的判别矩阵却比较复杂。在分析了两种充分条件具有不同判别矩阵的原因的基础上,推导了条件极值的充分条件的判别方法,并采用一阶泰勒展开求解自变量增量间的关系式,得到了高维多约束状态下条件极值充分条件的一种较精确的判别矩阵。有助于理解两种充分条件的关联及差别,提供了一种寻找精确的条件极值的充分条件的判别矩阵的方法。

The sufficient conditions for unconditional extreme values are determined according to Hessian matrix, but the sufficient conditions for conditional extreme values are rather complex. Based on an analysis that the identification of the two kinds of sufficient conditions rely on different matrices, we derived a method of locating the sufficient conditions for conditional extreme values. By use of first order Taylor expansion in obtaining the relations among argument increments, a more precise matrix for determining the sufficient condition for conditional extreme values in high dimensions and with multiple constraints has been deduced. This helps a better understanding of the relationships and differences between the two kinds of sufficient conditions, and provides a means of finding a more precise sufficient condition for a conditional extreme value.