Constrained optimization

Can one provide some intuition of how to solve the problem defined by Eqs. (28.7) and (28.8)? First, ignore the right hand side of Eq. (28.7) and recall that $z(\vec q)$ is just a scalar function in high dimensional space. If $\vec q$ had only two dimensions, then $z(\vec q)$ could be interpreted as a height function.

The task is to find the global minimum of this function. This is for example similar to finding a global maximum of a fitness function in evolutionary computing.

Since $z(\vec q)$ is analytically given, one can use mathematics to find candidates for global minima. As is known from calculus, all $\vec q^*$ where $\nabla z \, (\vec q^*) = \vec 0$ are such candidates. If the problem is constrained, additional candidates are along the boundaries of the allowed regions, see Fig. 28.3. A formal description of this leads to notions such as the Kuhn-Tucker-conditions and Lagrangian multipliers.

Figure 28.3: Constrained optimization