Theorie

Introduction

The Karush-Kuhn-Tucker (KKT) Theorem is a model of nonlinear optimization (NLP). The model is based on the Langrangian optimization, but considers inequality as part of the KKT constraints. The approach proofs the optimality of a (given) point concerning a nonlinear objective function. The satisfaction of KKT constraint is a necessary condition for a solution being optimal in NLP.

KKT Conditions

The six KKT condition are based on the Langrangian function of a random maximization (or minimization) problem.

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): L(x,\lambda) = f(x_1 , ... , x_n ) - \sum_{i=0} ^k \lambda_i * g_i (x_1,...,x_n) , where $f(x_{1},...,x_{n})$ is the objective function and $g_{i}(x_{1},...,x_{n})=0$ are the constraints. The algebraic sign of Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f(x_1,...,x_n)

depends on the state of the problem. For a maximization problem it is “+” for minimization “-“. The reason therefore is easy to see,when reflecting the objective function on the x-axis.The graphic below clarifies this coherence. It is visible, that only the algebraic sign of Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.):  \frac {\delta f} {\delta x} 
changes, because $\max f(x)$ = Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.):  \min –(f(x)) 

. The restrictions (in this case $x$ has to be lower than 1) don’t change.