Calculus Optimization Methods/Lagrange Multipliers

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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 [2].

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 .