How to find 3d point with minimum sum of euclidean distances to all given segments?

Let the segments be p₁ q₁, …, p_n q_n. We formulate an optimization problem:

minimize ∑_i ‖x − ((1 − y_i) p_i + y_i q_i)‖
subject to
x ∈ ³
∀i, y_i ∈ [0, 1]

The variable x is the point we’re looking for. The variables y_i take advantage that we’re trying to minimize a minimum (minimum distance to x) and are used to form the convex combination in the objective.

This is a convex problem, so either cvxpy or scipy.optimize should be able to handle it nicely.