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 [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 .