Definition:Lagrange's Method of Multipliers/Lagrange Multiplier
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Definition
Consider a real-valued function $\map f {x_1, x_2, \ldots, x_n}$ subject to one or more equality constraints $\map {g_i} {x_1, x_2, \ldots, x_n} = 0$.
Let Lagrange's method of multipliers be used to find maxima or minima of $f$ by minimizing:
- $L = f + \lambda_1 g_1 + \lambda_2 g_2 + \cdots$
with respect to the $x_i$ and $\lambda_i$.
The coefficients $\lambda_i$ are known as the Lagrange multipliers for this process.
Examples
Arbitrary Example
Find the maximum $M$ of the function $u: \R^2 \to \R$ defined as:
- $\forall \tuple {x, y} \in \R^2: \map u {x, y} = x y$
subject to the constraint:
- $\text C: \quad x + y = 1$
Also known as
A Lagrange multiplier is also known as an undetermined multiplier.
Also see
- Results about Lagrange's method of multipliers can be found here.
Source of Name
This entry was named for Joseph Louis Lagrange.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Lagrange multipliers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Lagrange multipliers