Using parameter elimination to solve discrete linear Chebyshev approximation problems

We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of approximation errors. Such problems find wide application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is extensively used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the approximation problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved that allows to handle the problem by reducing to a problem with one unknown parameter eliminated together with a box constraint imposed on this parameter. We apply this result to derive direct solutions of problems of low dimension, formulated as linear regression problems with one and two parameters. Furthermore, we develop a procedure to solve multidimensional approximation (multiple linear regression) problems. The procedure is based on a direct solution method that comprises two phases: backward elimination and forward determination (substitution) of the unknown parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two illustrative numerical examples.