Lagrange multipliers

Lagrange multipliers

The method of Lagrange multipliers is a method for finding an optimum in a function subject to some equality constraints. As an example this method can be used to find an optmum in the function $f(x,y)$ subject to the constraint $g(x,y)=a$. This method does so by introducing a new variable $\lambda$ called a Lagrange multiplier. The extended function $L(x,y,\lambda)$ below is then introduced: $$L(x,y,\lambda) = f(x,y) - (\lambda - a)g(x,y)$$ is then introduced. It can then be shown that the values of $x$ and $y$ in the unconstrained opimum for $L(x,y,\lambda)$ are the same as the values of $x$ and $y$ at the constrained optimum in $f(x,y)$

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