A Finite Newton Algorithm for Non-degenerate Piecewise Linear Systems

Xiao–Tong Yuan, Shuicheng Yan; JMLR W&CP 15:841-854, 2011.

Abstract

We investigate Newton-type optimization methods for solving piecewise linear systems (PLS) with non-degenerate coefficient matrix. Such systems arise, for example, from the numerical solution of linear complementarity problem which is useful to model several learning and optimization problems. In this paper, we propose an effective damped Newton method, namely PLS-DN, to find the exact solution of non-degenerate PLS. PLS-DN exhibits provable semi-iterative property, i.e., the algorithm converges globally to the exact solution in a finite number of iterations. The rate of convergence is shown to be at least linear before termination. We emphasize the applications of our method to modeling, from a novel perspective of PLS, several statistical learning problems such as elitist Lasso, non-negative least squares and support vector machines. Numerical results on synthetic and benchmark data sets are presented to demonstrate the effectiveness and efficiency of PLS-DN on these problems.

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