Nonlinear Opt.: KKT- theorem 5

Karush-Kuhn Tucker Theorie (KKT-Theorie) :

Karush-Kuhn Tucker Theorem (KKT-Theorem) is a mathematical Method to solve Nonlinear optimization problems with help of Lagrangian Function. It requires inequality constraints. The system of equations corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically.

Optimality conditions according to KKT-Theorem:

#Generalization of the Lagrangian condition #Take the inequality Constraints in consideration #Necessary (even sufficient under certain preconditions) conditions which shall be satisfied by an extremizer of the Lagrangian function.

Nonlinear minimization problem

Consider the nonlinear optimization Problem :

Min $z$ = $f$($x_{1}$, …., $x_{n}$)

s.t. $h_{i}$ ($x_{1}$, ...,$x_{n}$ ) – b ≤0 bzw. $g_{i}$($x_{1}$,. ..,$x_{n}$) ≤ 0 , $x_{j}$ ≥ 0

$i=1,...,m$  ; $j=1,...,n$.

f and g are continuously differentiable

Lagrangian Function for a optimization Problem who has to be minimized:

$L(x;\lambda )$ = $f$($x_{1}$,…,$x_{n}$)+ Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum_{k=1}^m\lambda_i

 $g_{i}$ ($x_{1}$,…,$x_{n}$)

Or

$L(x;\lambda )=f(x)+\lambda ^{T}g(x)$

A vector Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): (x ̂, \lambda ̂) in $R_{+}^{n+m}$ is called saddle point of Lagrangian function $L(x,\lambda )$, if it holds for all x ∈$R_{+}^{n}$ And $\lambda$ ∈$R_{+}^{m}$: