Calculus Optimization Methods/Lagrange Multipliers
The method of Lagrange multipliers solves the constrained optimization problem by transforming it into a non-constrained optimization problem of the form:
Then finding the gradient and hessian as was done above will determine any optimum values of .
Suppose we now want to find optimum values for subject to from .
Then the Lagrangian method will result in a non-constrained function.
The gradient for this new function is
Finding the stationary points of the above equations can be obtained from their matrix from.
This results in .
Next we can use the hessian as before to determine the type of this stationary point.
Since then the solution minimizes subject to with .