A Spline Smoothing Newton Method for L∞ Distance Regression with Bound Constraints

Orthogonal distance regression is arguably the most common criterion for fitting a model to data with errors in the observations. It is not appropriate to force the distances to be orthogonal, when angular information is available about the measured data points. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of l2 norm by l∞ norm. This criterion may be a more appropriate one in the context of accept/reject decisions for manufacture parts. For l∞ distance regression with bound constraints, we give a smoothing Newton method which uses cubic spline and aggregate function, to smooth max function. The main spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; hence it acts also as an active set technique, so it is more efficient for the problem with large amounts of measured data. Numerical tests in comparison to some other methods show that the new method is very efficient.