Minimax approximation from a least squares solution

One of the most popular algorithms for solving minimax approximation problems is the Barrodale and Phillips simplex-based method [I. Barrodale and C. Phillips (1974). An improved algorithm for discrete Chebyshev linear approximation. In: Proceedings of the Fourth Manitoba Conference on Numerical Mathematics, pp. 177-190, University of Manitoba, Winnipeg, Canada.]. Our new research has found that the modified pivoting strategy in the first stage of the Barrodale and Phillips algorithm could result in an invalid starting point and system failure in later stages [D. Lei (2002). Robust and efficient algorithms for ℓ1 and ℓ∞ approximations. PhD Thesis, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK.]. In contrast, a primal method that we propose in this paper is numerically more stable, and the approximating function does not suffer the effects of wild oscillation. However, on average, the primal method requires more searching steps to find an optimal solution. To improve the efficiency of the primal method, we have considered using the least squares solution to provide a better starting point. Experimental results show that if the least squares estimator is available, then it can speed up the convergence rate significantly in most cases. The number of steps required by using the least squares solution for the primal method also compares favourably with that of the dual method