A global optimization procedure for the location of a median line in the three-dimensional space

A global optimization procedure is proposed to find a line in the Euclidean three-dimensional space which minimizes the sum of distances to a given finite set of three-dimensional data points.Although we are using similar techniques as for location problems in two dimensions, it is shown that the problem becomes much harder to solve. However, a problem parameterization as well as lower bounds are suggested whereby we succeeded in solving medium-size instances in a reasonable amount of computing time.

► We solve the median line location problem in the three-dimensional space.
► A branch-and-bound algorithm is suggested and necessary lower bounds are calculated.
► Extensive numerical results show that medium-size instances can be solved.