Local minimum problem?

Question

Check this question:

http://answers.yahoo.com/question/index;_ylt=AiKobJxCAaRYm8wci.6glo_sy6IX;_ylv=3?qid=20071103182545AA1l69b&show=7#profile-info-d80dc8bc16566bd4b6363642fd3d9034aa

Take the simpler 2D problem of finding minimum total sum of distances from a point on the plane to n other points. A local minimum is a point where this total sum of distances to the other n points is a minimum only locally about the point, but not necessarily minimum for the entire plane. Do such local minimums exist other than the one that's the minimum for the entire plane? If yes, can you give an example, if not, can you show why?

As smci mentions, in the case of n = 3, this minimum is called the Fermat Point.

Answer 1

Let me see if I understand this correctly.

Given a set of position vectors N, and a point (x, y), you want to minimize

f(x,y) = Σ ||Ni - (x, y)||.

By hypothesis, such a minimum exists, though it may or may not be unique....I'll have to think about it.....

Calculus probably won't help here, you'd have to use a computer for a large number of points.

The Nelder-Mead Algorithm may be applicable to this problem......
http://math.fullerton.edu/mathews/n2003/NelderMeadMod.html