# Statistical Thermodynamics and Rate Theories/Lagrange multipliers

## Constrained Optimization[edit]

When constraints are imposed on a system, Lagrange multipliers can be incorporated to determine the maximum of the multivariable function. The new process for determining maximum of this function within the constraint becomes:

- Write the constraint as a function. i.e.,
- Define a new equation. where is an undefined constant
- Solve this set of equations to find the maximum, using the previous three steps for determining the maximum of an unconstrained system.

For example, suppose we have a function and we impose the following constraint upon the function:

The constraint would be written as

We would then define the new equation following the constraint as

Next, we take the partial derivative with respect for both x and y, set it to zero, and solve for x and y.

First of all taking the partial derivative with respect for x set to zero:

Evaluating this will give:

Now, the partial derivative with respect for y set to zero will be taken:

This leads to the following system of equations that can be solved to determine the maximum:

From Solving this set of equations the maximum subjected to the constraint of is found to be:

## Unconstrained Optimization[edit]

Determining the maximum of an unconstrained system follows very similar steps it just will not have a lagrange multiplier as the system is not subjected to the

maximum along a given line, rather the maximum of the system itself. The steps to solve an unconstrained system becomes:

- Calculate the partial derivatives
- set them to zero
- solve for the variables

For example given the same function of first the partial derivatives will be calculated to be:

Let both partial derivatives be zero:

Finally the variables will be solved for. Giving a maximum that is different than that of the constrained system: