What are Lagrange multipliers?

Lagrange multipliers tend to find the maximum or minimum values of a function. In this Answer, we'll first look at the Lagrange multiplier theorem and then move on to a practical example where such multipliers come in handy.

The Lagrange multiplier theorem

The problem we want to optimize has the functions $f(x,y,z)$and $l(x,y,z)$

The latter function is equal to a constant $k$.

The Lagrange multipliers become the coefficients of the linear combination of gradients. They are often denoted by $\lambda$ and their values are found by taking into account both functions and applying the constraints given.

A simple syntax for the problem is as follows:

Now, let's move on to an example where we have to solve a problem using Lagrange multipliers.

Example

Let's suppose that we have two equations given as follows:

We are supposed to use the Lagrange multipliers to find either the maximum or minimum values. To do that we will apply $\lambda$ to our $eq.2$ variables one by one and make them equal to $eq.1$:

Putting values into the above equations will give us:

We can now substitute the variables $x$, $y$, and $z$ with their respective $\lambda$ values in the equation $eq.2$:

We can use the rounded value of $\lambda$ to find the values of $x$, $y$ and $z$ as follows:

Hence, we can now say that $f(1.5,3,4.5)$has the local minimum value. To prove this, we will find the value of $f$ on these values and then use two more sets of $x,y,z$ to compare their values to the minima:

Let's take two more sets as follows:

Hence, you can see that the Lagrange multipliers provide us with the minima$58.5$. There is no other set of $x,y,z$ that can give a local minimum lower than this value.

Note: Notice that the sets chosen have to fulfill the constraint defined in $eq.2$.

Conclusion

To maximize or minimize a multi-variate function:

1. Introduce a variable $\lambda$ and define function $L$ as follows:

2. Set the gradient of $L$ equal to zero to find the critical points:

3. Remove the $\lambda$ and plug it into the function $f$. Whichever gives the greatest or smallest value is the maximum