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