$\mathbf{p}$ is a vector with dimension: $x \times 1$
$\mathbf{d}$ is a vector with dimension: $1 \times y$
$\mathbf{V}$ is a matrix with dimension: $x \times y$
$y \geq x$
- $\mathbf{d}$ and $\mathbf{V}$ are known.
Aaaaand: $$ \mathbf{d}_{1y} = \sum ^{x}_{i=1} \left( \mathbf{p}_{i1}-\mathbf{V}_{iy} \right )^2 $$
(I'm not sure if the subscript '1' is required for $\mathbf{d}$ and $\mathbf{p}$, please ignore them if it's not the usual notation)
I need to solve for $\mathbf{p}$.
I've tried to put the vectors into the same dimension as $\mathbf{V}$ to see if I could use algebra to nut it out, but the square in particular made we wary of the result, (let alone the summation).
How is $\mathbf{p}$ pulled out of the summation and square? (i.e. how do I solve for $\mathbf{p}$?)
The matrix $V$ can be written as $y$ column vectors $v_j$, each of dimension $x$: $$ V = (v_1, \dotsc, v_y) $$ Then the $j$-th component of vector $d$ is the scalar product of the difference vector $p-v_j$ with itself, which is its Euclidean norm squared: $$ d_j = (p - v_j)^2 = \lVert p - v_j \rVert_2^2 = \sum_{i=1}^x (p_i - v_{ij})^2 = r_j^2 \quad (j \in \{1,\dotsc,y\}) $$ These are spheres with center at vector $v_j$ and radius $r_j = \sqrt{d_j}$.
So $p$ is an element of the intersection of the above $y$ spheres in $x$ dimensions.
This seems to be a generalization of the trilateration problem.
Example:
$$ V = \begin{pmatrix} 1 & 3\\ 2 & 4 \end{pmatrix} \quad d = \begin{pmatrix} 1 \\ 4 \end{pmatrix} $$
Then $$ d_1 = 1 = (p_1 - 1)^2 + (p_2 - 2)^2 \\ d_2 = 4 = (p_1 - 3)^2 + (p_2 - 4)^2 $$
The circles in 2D can have zero, one, two or infinite many intersection points, where the last case is two congruent circles.
In 3D we have zero or one intersection points or a circle of intersection or congruence.
In higher dimensions I do not know the cases.
Formulation as Non-Linear Least Squares Problem:
In presence of errors one define the residues $$ e_j(p) = d_j - \sum_{i=1}^x (p_i - v_{ij})^2 $$ and the task is to find a vector $p$ which minimizes $$ S(p) = e^2 = \sum_{j=1}^y e_i^2 $$ This seems to be a non-linear least squares problem.