Regularized Nonsmooth Newton Algorithms for Best Approximation

Yair Censor; Walaa M. Moursi; Tyler Weames; Henry Wolkowicz

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arXiv · 2212.13182

Regularized Nonsmooth Newton Algorithms for Best Approximation

Yair Censor·Walaa M. Moursi·Tyler Weames·Henry Wolkowicz

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Abstract

We consider the problem of finding the best approximation point from a polyhedral set, and its applications, in particular to solving large-scale linear programs. The classical projection problem has many various and many applications. We study a regularized nonsmooth Newton type solution method where the Jacobian is singular; and we compare the computational performance to that of the classical projection method of Halperin-Lions-Wittmann-Bauschke (HLWB). We observe empirically that the regularized nonsmooth method significantly outperforms the HLWB method. However, the HLWB has a convergence guarantee while the nonsmooth method is not monotonic and does not guarantee convergence due in part to singularity of the generalized Jacobian. Our application to solving large-scale linear programs uses a parametrized projection problem. This leads to a \emph{stepping stone external path following} algorithm. Other applications are finding triangles from branch and bound methods, and generalized constrained linear least squares. We include scaling methods that improve the efficiency and robustness.

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Yair Censor, Walaa M. Moursi, Tyler Weames, Henry Wolkowicz. 2022-12-26. Regularized Nonsmooth Newton Algorithms for Best Approximation. https://arxiv.org/abs/2212.13182

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