Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange.

The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as

$${\displaystyle {\mathcal {L}}(x,\lambda )\equiv f(x)+\langle \lambda ,g(x)\rangle }$$

for functions ${\displaystyle f,g}$; the notation ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes an inner product. The value ${\displaystyle \lambda }$ is called the Lagrange multiplier.

In simple cases, where the inner product is defined as the dot product, the Lagrangian is

$${\displaystyle {\mathcal {L}}(x,\lambda )\equiv f(x)+\lambda \cdot g(x)}$$

The method can be summarized as follows: in order to find the maximum or minimum of a function ${\displaystyle f}$ subject to the equality constraint ${\displaystyle g(x)=0}$, find the stationary points of ${\displaystyle {\mathcal {L}}}$ considered as a function of ${\displaystyle x}$ and the Lagrange multiplier ${\displaystyle \lambda ~}$. This means that all partial derivatives should be zero, including the partial derivative with respect to ${\displaystyle \lambda ~}$.

or equivalently