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In this paper, estimation of a regression function from independent and identically distributed random variables is considered. Estimates are defined by minimization of the empirical [L.sub.2] risk over a class of functions, which are defined as maxima of minima of linear functions. Results concerning the rate of convergence of the estimates are derived. In particular, it is shown that for smooth regression functions satisfying the assumption of single index models, the estimate is able to achieve (up to some logarithmic factor) the corresponding optimal one-dimensional rate of convergence. Hence, under these assumptions, the estimate is able to circumvent the so-called curse of dimensionality. The small sample behavior of the estimates is illustrated by applying them to simulated data. Index Terms--Adaptation, dimension reduction, [L.sub.2] error, nonparametric regression, rate of convergence, single index model.

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2004.06.014 Byline: Adil M. Bagirov, John Yearwood Keywords: Nonsmooth optimization; Cluster analysis; Minimum sum-of-squares clustering Abstract: The minimum sum-of-squares clustering problem is formulated as a problem of nonsmooth, nonconvex optimization, and an algorithm for solving the former problem based on nonsmooth optimization techniques is developed. The issue of applying this algorithm to large data sets is discussed. Results of numerical experiments have been presented which demonstrate the effectiveness of the proposed algorithm. Author Affiliation: Centre for Informatics and Applied Optimization, School of Information Technology and Mathematical Sciences, University of Ballarat, P.O. Box 663, Vic. 3353, Australia Article History: Received 25 November 2002; Accepted 17 June 2004

Piecewise linear L 1 -regression problem is formulated as an unconstrained difference of convex (DC) optimization problem and an algorithm for solving this problem is developed. Auxiliary problems are introduced to design an adaptive approach to generate a suitable piecewise linear regression model and starting points for solving the underlying DC optimization problems. The performance of the proposed algorithm as both approximation and prediction tool is evaluated using synthetic and real-world data sets containing outliers. It is also compared with mainstream machine learning regression algorithms using various performance measures. Results demonstrate that the new algorithm is robust to outliers and in general, provides better predictions than the other alternative regression algorithms for most data sets used in the numerical experiments. [ABSTRACT FROM AUTHOR]