Engineering Analysis/Minimization

Khun-Tucker Theorem

The Khun-Tucker Theorem is a method for minimizing a function f(x) under the constraint g(x). We can define the theorem as follows:

${\displaystyle L(x)=f(x)+\langle \Lambda ,g(x)\rangle }$

Where Λ is the lagrangian vector, and < , > denotes the scalar product operation. We will discuss scalar products more later. If we differentiate this equation with respect to x first, and then with respect to Λ, we get the following two equations:

${\displaystyle {\frac {\partial L(x)}{\partial x}}=x+A\Lambda }$
${\displaystyle {\frac {\partial L(x)}{\partial \Lambda }}=Ax-b}$

We have the final result:

${\displaystyle x=A^{T}[AA^{T}]^{-1}b}$