Fitting parametric curves and surfaces by l∞ distance regression

For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of least squares by the Chebyshev norm. For example, this criterion may be a more appropriate one in the context of accept/reject decisions for manufactured parts. The resulting problem has some interesting features: it has much structure which can be exploited, but generally the solution is not unique. We consider a method of Gauss-Newton type and show that if the non-uniqueness is resolved in a way which is consistent with a particular way of exploiting the structure in the linear subproblem, this can not only allow the method to be properly defined, but can permit a second order rate of convergence. Numerical examples are given to illustrate this.