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 norm by norm. This criterion may be a more appropriate one in the context of accept/reject decisions for manufacture parts. For 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. 1. Introduction For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression (ODR). Given the data pairs , , where is the independent variable and is the dependent variable; suppose where is a vector of parameters to be determined. We assumed that is the random error associated with and that is the random error associated with . To be precise, we relate the quantities , , , and to As shown in Boggs et al. [1] this gives rise to the ODR problem given by The ODR problem can be solved by the Gauss-Newton or Levenberg-Marquardt methods (see [1, 2]). The general form of the bounded constrained ODR problem can be expressed by where and are vectors of length that provide the lower and upper bounds on , respectively. Zwolak et al. give the algorithm to handle (4) in [3]. It is not appropriate to force the distances to be orthogonal, when angular information is available about the measured data points, such as the rotated cone fitting problem and rotated paraboloid fitting problem. Then, (4) becomes where , ？ is an orthogonal matrix, , , , , and and are vectors of length . When the least squares norm is not appropriate, problem (5) can be generalized to use other measures in a variety of ways. Most generalizations have been of formulations (5), with the norms replaced with other norms. We consider here norms. It may be a more appropriate one in the context of accept/reject decisions for manufacture parts (see [4]). In this paper, we consider the following distance regression with bound constraints. Let
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