Nonlinear Opt.: Lagrangian condition 4

Theory

The Lagrangian method is part of the non-linear optiization theory. It is used to solve an objective function with constraints. The application area of this mothed is very wide-ranging.

Problem

Objective function $f(x_{1},...,x_{n})=min!\quad or\quad max!$

subject to $g_{k}(x_{1},...,x_{n})=c$ e element R

The objective function $f(x_{1},...,x_{n})$ and $g_{k}(x_{1},...,x_{n})$ must be continuosly differentiable!

Procedure

First you have to rewrite the constraints to the form of Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): g_k(x_1,...,x_n)-c = 0

Then you set up the Lagrangian function: $L(x_{1},...,x_{n};\lambda _{1},...,\lambda _{k}=f(x_{1},...,x_{n})+\lambda _{1}g_{1}(x_{1},...,x_{n})+...+\lambda _{k}g_{k}(x_{1},...,x_{n}$

Third you must derive the Lagrangian function with respect to all variables and set These derivatives to Zero .

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial}{\partial x_i}L(x_1,...,x_n; \lambda_1,...,\lambda_k = 0

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial}{\partial \lambda_k}L(x_1,...,x_n;\lambda_1,...,\lambda_m

Last you have to solve all the equations.