Assume I have a matrix $(d_{ji})_{ij}$ of distances between points $i$ and $j$. These distances could be anything fulfilling the triangle inequality.
Now I would like to find coordinates $(x_i,y_i)$ for each $i$, so that the Euclidean distances are always less or equal to the real ones:
$$ \varepsilon_{ij} = d_{ij} - \sqrt{(x_i-x_j)^2 + (y_i-y_j)^2} \geq 0$$
Furthermore, the sum $\sum_{ij} \varepsilon_{ij}$ should be minimal.
How could one approach such a non-linear problem?
EDIT: I would also be interested in other, similar objective functions that somehow minimize the $\varepsilon_{ij}$. The important for me is that the "approximation" fulfils the inequalities stated above.