“Lagrange Multiplier”的概念、定义、习题解题方法及翻译-数学辞典

单词

Lagrange Multiplier

释义

Lagrange Multiplier

Used to find the Extremum of subject to the constraint ,where and are functions with continuous first Partial Derivatives on the OpenSet containing the curve , and at any point on the curve(where is the Gradient). For an Extremum to exist,

(1)

But we also have

(2)

Now multiply (2) by the as yet undetermined parameter

and add to (1),

(3)

Note that the differentials are all independent, so we can set any combination equal to 0, and the remainder must stillgive zero. This requires that

(4)

for all

, ...,

. The constant

is called the Lagrange multiplier. For multiple constraints,

, ...,

(5)

See also Kuhn-Tucker Theorem

References

Arfken, G. ``Lagrange Multipliers.'' §17.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 945-950, 1985.

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