How Does the Lagrange-Newton (SOLVER) Method Work for Numerical Optimization?

[brydustin]

I have the analytical first and second derivatives of a (multidimensional) lagrangian ( l = f - λh). X is the vector of variables of the objective function and λ is the single lagrange multiplier.
where f=f(X) is the nonlinear objective function, h is the nonlinear (equality) constraint (i.e. h(X) - ρ = 0 at optimized solution). I'm generally confused about how to solve this (i've read about the "Lagrange-Newton (SOLVER) method" but don't really understand it.

How do I update X and λ? Please try to be as specific as possible. Thanks.

 

[gtoguy97]

The Lagrange-Newton (SOLVER) method is a numerical optimization technique that involves iteratively updating the variables X and λ in order to find an optimal solution. The algorithm involves computing the analytical first and second derivatives of the lagrangian (l) with respect to both X and λ, and then using the derivatives to construct the Newton-Raphson iteration equations. The update equations for X and λ are given by:X = X - [Hessian(l)]^-1 * Gradient(l)λ = λ + h(X) - ρ Where the Hessian matrix is the matrix of second derivatives of l with respect to X, and the gradient vector is the vector of first derivatives of l with respect to X. These update equations can then be iteratively applied until the objective function reaches its optimal value or the constraint is satisfied.