Engineering Analysis/Minimization

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Khun-Tucker Theorem[edit]

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