Finding a vector in Euclidian space that minimizes a loss function subject to some constraints I'm trying to solve the following minimization problem, and I'm sure there must be a standard methodology that I could use, but so far I couldn't find any good references. Please let me know if you have anything in mind that could help or any references that you think would be useful for tackling this problem.
Suppose you are given $K$ points, $p_i \in R^n$, for $i \in \{1,\ldots,K\}$. Assume also that we are given $K$ constants $\delta_i$, for $i \in \{1,\ldots,K\}$. We want to find the vector $x$ that minimizes:
$\min_{x \in R^n} \sum_{i=1,\ldots,K} || x - p_i ||^2$
subject the following $K$ constraints:
$\frac{ || x - p_i ||^2 } { \sum_{j=1,\ldots,K} ||x - p_j||^2} = \delta_i$
for all $i \in {1,\ldots,K}$.
Any help is extremely welcome!
Bruno
edit: also, we know that $\sum_{i=1,\ldots,K} \delta_i = 1$.

You can try using Lagrange multiplier method - see wikipedia.
