Easy transportation-like problems on K-dimensional arrays (Q1823138)

The K-dimensional version of two transportation-like problems is posed and efficiently solved. The case \(K=2\) goes as follows: Given an (m,n)- matrix A of reals, a real m-vector u, and a real n-vector v, find a real (m,n) matrix X minimizing \[ either\quad f(X)=\sum^{m}_{i=1}\sum^{n}_{j=1}(X_{ij}-A_{ij})^ 2\quad or\quad g(X)=\max_{ij}\{X_{ij}-A_{ij}\}, \] subject to linear constraints of transportation type, i.e., \[ \sum^{n}_{j=1}X_{ij}=u_ i,\quad i=1,...m,\quad \sum^{m}_{i=1}X_{ij}=v_ j,\quad j=1,...,n. \] In the generalization for any \(K\geq 2\), consideration of the f(X)-type of objective function yields an explicit formula for the optimal solution, whereas for the g(X)-type a one-pass algorithm is proposed.