View source for Engineering Analysis/Minimization

{{Engineering Analysis}}
{{info|Before reading this chapter, the student should know what ''minimization'' is, and how to minimize a function. Students should also know ''partial differentiation'', and how to solve systems of equations.}}
== 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:

:<math>L(x) = f(x) + \langle \Lambda, g(x)\rangle</math>

Where &Lambda; 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 &Lambda;, we get the following two equations:

:<math>\frac{\partial L(x)}{\partial x} = x + A\Lambda</math>

:<math>\frac{\partial L(x)}{\partial \Lambda} = Ax - b</math>

We have the final result:

:<math>x = A^T[AA^T]^{-1}b</math>

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