Let $A$ and $B$ be two sets of $d$-dimensional coordinates of comparible (though not necessarily equal length). For example, here we can set $d = 3$ and have normalized values like:
A = {{0.446362, 0.811292, 0.0922594},{0.244159, 0.260142, 0.304582}, {0.122547, 0.269019, 0.0644819},
{0.448114, 0.828661, 0.733406},{0.347111, 0.296056, 0.398861}, {0.219952, 0.653454, 0.09107},
{0.592839, 0.0333978, 0.210341},{0.0970157, 0.132429, 0.366429}, {0.294297, 0.527056, 0.403706},
{0.196355, 0.115887, 0.636247},{0.993047, 0.51482, 0.229379}, {0.251266, 0.135959, 0.178469},
{0.789068, 0.182785, 0.338934}, {0.42705, 0.151849, 0.671407}, {0.796922, 0.229053, 0.946328},
{0.292144, 0.148314, 0.116313}, {0.589524, 0.474425, 0.0958731}, {0.181622, 0.140052, 0.958391},
{0.0376616, 0.614436, 0.394617}};
B = {{0.690322, 1.17215, 0.961834}, {0.271458, 0.942832, 0.745714}, {0.508862, 1.03457, 0.3475},
{0.81515, 0.378891, 0.465227}, {0.6036, 1.34974, 0.426426}, {0.405347, 0.463069, 1.27554},
{1.12031, 0.604772, 1.19204}, {0.373017, 0.343105, 0.595875}, {0.642332, 0.602587, 0.678469},
{0.884344, 0.746337, 0.38915}, {1.22961, 0.47189, 0.712069}, {0.56058, 0.755362, 0.648201},
{0.596699, 0.465542, 0.646428}, {0.48087, 0.522362, 0.315634}, {0.752854, 1.17231, 0.421088},
{0.693156, 0.546617, 0.9851}, {0.507197, 0.426626, 0.847138}, {0.584263, 0.452749, 0.422986},
{0.545495, 0.427736, 0.494085}};
(Note that the order of the elements in $A$ and $B$ are of no consequence.)
While a bijection isn't possible, I would like to find the best possible "rigid mapping" between the elements in $A$ and $B$ where we add some fixed real number vector $Q=(q_1,q_2,q_3)$ to each coordinate in $B$ to minimize a weighted distance between pairs of coordinates in $A$ and $B$ according to some axis-specific cost function. Consider that in the above example, I've rigged things to make $Q = (q_1,q_2,q_3) \approx (0.3,0.3,0.3)$ about optimal (the largest error is of magnitude $\approx 0.1$).
To specify this cost function, we select array $A$ if it has more points than $B$ and vice versa. For convenience, let's say this array is $A$. Now let $(d_{(1,i)},d_{(2,i)},d_{(3,i)})$ be the shortest distance between a point $b_i \in B$ and $a_j \in A$. I'd like to have a cost function that looks something like:
$C = \sum_{i=1}^{||B||} || ((d_{(1,i)})^{z_1},(d_{(2,i)})^{z_2},(d_{(3,i)})^{z_3}) ||$
Where $(z_1,z_2,z_3)$ represents an arbitrary set of exponents. For a single real numbered value cost, we could compute the Norm of the above sum.