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Iterative solution of piecewise linear systems for the numerical solution of obstacle problems
"... Abstract: We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the numerical solution of freesurface problems. In p ..."
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Abstract: We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the numerical solution of freesurface problems. In particular, we here study their application to the numerical solution of both the (linear) parabolic obstacle problem and the obstacle problem. We propose a class of effective semiiterative Newtontype methods to find the exact solution of such piecewise linear systems. We prove that the semiiterative Newtontype methods have a global monotonic convergence property, i.e., the iterates converge monotonically to the exact solution in a finite number of steps. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.
with application to positively constrained convex
, 2015
"... A semismooth Newton method for a special piecewise linear system ..."
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1Nondegenerate Piecewise Linear Systems: A Finite Newton Algorithm and Applications in Machine Learning
"... We investigate Newtontype optimization methods for solving piecewise linear systems (PLSs) with nondegenerate 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 ..."
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We investigate Newtontype optimization methods for solving piecewise linear systems (PLSs) with nondegenerate 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 PLSDN, to find the exact (up to machine precision) solution of nondegenerate PLSs. PLSDN exhibits provable semiiterative 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 in modeling, from a novel perspective of PLSs, some statistical learning problems such as box constrained least squares, elitist Lasso (Kowalski & Torreesani, 2008) and support vector machines (Cortes & Vapnik, 1995). Numerical results on synthetic and benchmark data sets are presented to demonstrate the effectiveness and efficiency of PLSDN on these problems. 1