Theorie
Karush-Kuhn-Tucker Theorem
Inhaltsverzeichnis
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 is the objective function and are the constraints.
The algebraic sign of Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum_{i=0} ^k \lambda_i * g_i (x_1,...,x_n)
depends on the state of the problem, for a maximization problem it is “-” for minimization “+“.
The first KKT Condition is the derivative of after all , which is equal to the gradient of , Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \nabla L (x,\lambda)
and it includes the inequality:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 1) \frac {\partial L}{\partial x_i} = \frac {\partial f}{\partial x _i} - \sum_{i=0} ^j * \frac {\partial g_i}{\partial x_i} \leq 0
If the objective function has to be minimized instead, only the sign of Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac {\partial f}{\partial x_i}
changes, because = Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): -(f(x))
. The reason therefore is easy to see,when reflecting the objective function on the abcissa,which is clarified by the graphic below. The restrictions (in this case Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x \le 1
) don’t change.
The second KKT constraint ensures, any x can not be less than 0, which is basically the non-negativity constraint.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 2) x_i * \frac {\partial L}{\partial x_i} = 0
The third and fourth constraints are analog to the first and the second one, but this time for .
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 3) \frac {\partial L} {\partial \lambda_j} \leq 0
Analogue to the previous changes when one face a minimization problem, the sign of changes.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 4) \lambda_j * \frac {\partial L}{\partial \lambda_j} = 0
And finally the non-negativity constraints:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 5) x_i \geq 0 , i=1,...,n
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 6) \lambda_j \geq 0 , j = 1,...,k
Examples
Further Information
If is concave and all are convex, the satisfaction of all KKT conditions is necessary and sufficient for a global optimum, if the Slater condition is satisfied to. This means, there exist an Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x \in \R_{+}^n , so that which means there is a point inside the solution space. Node that is the original constraint, but to make things easy those restrictions are changed from inequality functions into quality functions ( ).