This paper is concerned with the computation of solutions to the problem of best near-interpolation by parametric curves to constraints of the form f(ti) ∈ Ki for general sets Ki , including derivative constraints when ti has multiplicity greater than one. The minimizers are spline curves that we represent as smoothing splines with weights that are determined from the Lagrange multipliers corresponding to the constraints. To compute approximate solutions, we generalize a particular fixed point iteration used previously in nearinterpolation; to compute “exact” solutions, we apply a non-smooth Newton solver. The construction leads to optimality conditions for near-interpolation with variable data sites for these more general constraints. Standard methods are used to update these data sites.