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Lagrange Multiplier


Lagrange multipliers, also called Lagrangian multipliers (e.g., Arfken 1985, p. 945), can be used to find the extrema of a multivariate function f(x_1,x_2,...,x_n) subject to the constraint g(x_1,x_2,...,x_n)=0, where f and g are functions with continuous first partial derivatives on the open set containing the curve g(x_1,x_2,...,x_n)=0, and del g!=0 at any point on the curve (where del is the gradient).

LagrangeMultipliers

For an extremum of f to exist on g, the gradient of f must line up with the gradient of g. In the illustration above, f is shown in red, g in blue, and the intersection of f and g is indicated in light blue. The gradient is a horizontal vector (i.e., it has no z-component) that shows the direction that the function increases; for g it is perpendicular to the curve, which is a straight line in this case. If the two gradients are in the same direction, then one is a multiple (-lambda) of the other, so

 del f=-lambdadel g.
(1)

The two vectors are equal, so all of their components are as well, giving

 (partialf)/(partialx_k)+lambda(partialg)/(partialx_k)=0
(2)

for all k=1, ..., n, where the constant lambda is called the Lagrange multiplier.

The extremum is then found by solving the n+1 equations in n+1 unknowns, which is done without inverting g, which is why Lagrange multipliers can be so useful.

For multiple constraints g_1=0, g_2=0, ...,

 del f+lambda_1del g_1+lambda_2del g_2+...=0.
(3)

See also

Kuhn-Tucker Theorem

Portions of this entry contributed by David Gluss

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References

Arfken, G. "Lagrange Multipliers." §17.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 945-950, 1985.Lang, S. Calculus of Several Variables. Reading, MA: Addison-Wesley, p. 140, 1973.Simmons, G. F. Differential Equations. New York: McGraw-Hill, p. 367, 1972.Zwillinger, D. (Ed.). "Lagrange Multipliers." §5.1.8.1 in CRC Standard Mathematical Tables and Formulae, 31st Ed. Boca Raton, FL: CRC Press, pp. 389-390, 2003.

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Lagrange Multiplier

Cite this as:

Gluss, David and Weisstein, Eric W. "Lagrange Multiplier." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LagrangeMultiplier.html

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