Definition:Lagrange's Method of Multipliers

Definition

Lagrange's method of multipliers is a technique for finding maxima or minima of a real-valued function $\map f {x_1, x_2, \ldots, x_n}$ subject to one or more equality constraints $\map {g_i} {x_1, x_2, \ldots, x_n} = 0$.

The solution is found by minimizing:

$L = f + \lambda_1 g_1 + \lambda_2 g_2 + \cdots$

with respect to the $x_i$ and $\lambda_i$.

Lagrange Multipliers

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

Lagrange's method of multipliers is also known as just Lagrange's method.

Some sources style it as the Lagrange method of multipliers.

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