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:

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

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:

${\frac {\partial L(x)}{\partial x}}=x+A\Lambda$

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

We have the final result:

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