I have the following problem: $$ \arg\min_{\{x_i\}}\sum_{i,j}w_{i,j}\left\Vert x_{i}-c_{j}\right\Vert ^{2},\\ s.t. x_i^Tx_i=1\forall i,\quad x_i\succeq 0\forall i $$
where $x_i\in \mathbb R^K$, $i=1,...,M$ and $c_i\in \mathbb R^K$, $i=1,...,N$, are vectors, and $w_{i,j}$ is a scalar.
That is, I would like to find a set of suitable vectors $\{x_i\}_{i=1}^M$ that minimize the objective, constrained on the positive part of the unit sphere. Appreciate any directions...