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Regularized Nonsmooth Newton Algorithms for Best Approximation

Yair CensorWalaa M. MoursiTyler WeamesHenry Wolkowicz
Dec 2022
摘要
We consider the problem of finding the best approximation point from apolyhedral set, and its applications, in particular to solving large-scalelinear programs. The classical projection problem has many various and manyapplications. We study a regularized nonsmooth Newton type solution methodwhere the Jacobian is singular; and we compare the computational performance tothat of the classical projection method of Halperin-Lions-Wittmann-Bauschke(HLWB). We observe empirically that the regularized nonsmooth method significantlyoutperforms the HLWB method. However, the HLWB has a convergence guaranteewhile the nonsmooth method is not monotonic and does not guarantee convergencedue in part to singularity of the generalized Jacobian. Our application to solving large-scale linear programs uses a parametrizedprojection problem. This leads to a \emph{stepping stone external pathfollowing} algorithm. Other applications are finding triangles from branch andbound methods, and generalized constrained linear least squares. We includescaling methods that improve the efficiency and robustness.
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