A nearly ubiquitous problem in all types of
optimization schemes involving simultaneous optimization of many parameters is an effective way to find the global minimum of a system. Finding a
local minimum is relatively easy. A local minimum can be characterized as any point in space where traveling incrementally along any of the parameters results in a greater value of the function. There is often no practical way to determine whether a local minimum is the global minimum. The only way to be sure is to sample
all of parameter space. In the case of many parameters, this can be an impossible task.
A common situation where local minima may be a problem is in fitting a set of data to a model curve. Using educated starting guesses and sophisticated nonlinear least squares algorithms for minimizing the difference between the data and the model, an acceptable fit may be realized. The minimum is usually detected by measuring the goodness of fit with some statistic like the weighted residuals or chi2.
In computational modelling, particularly molecular dynamics, finding the global minimum is also a very very difficult problem. Theoretically, the native state of a protein is supposed to be the energetic global minimum of all the possible conformations a protein may assume. If you take an extended protein polymer and write a program that comes up with a folded protein, how do you know you have found the most stable structure? The minimum found may just be a metastable state. The problem for computational models is effective sampling. At what point have you explored a sufficient number of conformations that you are assured the one you have found is the best. There is no clear answer to this question.
There are cases when even nature gets stuck in a local minimum while looking for a global one. This is the origin of prion mediated diseases (see writeup on prions). The kinetic minimum is the local helical minimum the protein is trapped in. Only through a catalyzed conformational change caused by another prion can the global beta-sheet minimum be found.
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