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Projection onto simplicial cones by a semi-smooth Newton method

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Abstract

In this paper the problem of projection onto a simplicial cone is studied. By using Moreau’s decomposition theorem for projecting onto closed convex cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone is always well defined, and the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented. It is also shown that under a somewhat restrictive assumption, the semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone has finite convergence. Besides, under a mild assumption on the simplicial cone, the generated sequence converges linearly to the solution of the associated system of equations.

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Notes

  1. see the popularity of the Wikimization page Projection on Polyhedral Cone at http://www.convexoptimization.com/wikimization/index.php/Special:Popularpages.

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Acknowledgments

The first author was supported in part by FAPEG, CNPq Grants 471815/2012-8, 303732/2011-3 and PRONEX–Optimization(FAPERJ/CNPq). The authors wish to express their gratitude to the reviewers for their helpful comments and for a shorter, more elegant proof of Lemma 5.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, IME/UFG, Campus II- Caixa Postal 131, Goiânia, GO, 74001-970, Brazil

    O. P. Ferreira

  2. School of Mathematics, The University of Birmingham, The Watson Building, Edgbaston, Birmingham, B15 2TT, UK

    S. Z. Németh

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Ferreira, O.P., Németh, S.Z. Projection onto simplicial cones by a semi-smooth Newton method. Optim Lett 9, 731–741 (2015). https://doi.org/10.1007/s11590-014-0775-1

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