Quote:
Originally Posted by ShiningArcanine
Would someone elaborate on the nature of problems where the extreme value theorem cannot be used for minimization of a system of variables?

My understanding is that the complexity of the minimization job depends on how accurately you describe the forces involved, and whether subatomic forces are included in that.
Your book says that because the common wisdom holds that using gradientor Hessianbased minimization is impractical when there are tens of thousands of variables that all contribute to the minimum. Just calculating the forces that every atom exerts on all the others is a big job, and there is a large research literature on doing the computation efficiently. Finally, if you're looking for a global minimum somewhere inside a space that is very full of local minima, then you may never find it without massively sampling the entire space for the correct set of initial conditions needed. Certainly nobody expects a nonnumercial solution to such large problems, at least without simplifying away most of the detail that makes the problem relevant.
But I could be wrong, so ask your lab anyway.