无约束极值条件的一个证法

Proof for the unconstrained extremum conditions

无约束极值条件是最优化理论的重要组成部分，在理论上和实践上都有重要意义，约束问题的最优性条件是它的逻辑推广，解约束最优化问题的一种策略是解一系列无约束问题。本文首先介绍Ｒａｙｌｅｉｇｈ商定理，然后利用Ｒａｙｌｅｉｇｈ商定理给出无约束极值条件新的严格证明。这种方法比已有证法更简明，对那些不熟悉序列极限理论的读者，在学习和掌握最优化理论方面，有一定的实际意义。

The unconstrained extremum conditions are the important parts of optimization theory, and have certain significance in the theory and practice. The optimality conditions for constrained problems become a logical extension of the conditions for unconstrained problems. One strategy for solving a constrained problem is to solve a sequence of unconstrained problems. In this paper, the theorem of Rayleigh quotient is introduced at first, then by use of it, a new rigorous proof for an unconstrained extremum condition is given. This proof is more concise than the old one, and it has considerably practical value in learning and grasping the optimization theory for those readers who are not familiar with the theory of limit of sequence.