Optimization Letters

, Volume 9, Issue 4, pp 731–741 | Cite as

Projection onto simplicial cones by a semi-smooth Newton method

Original Paper


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.


Metric projection Simplicial cones Moreau’s decomposition theorem Semi-smooth Newton method 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Mathematics and StatisticsGoiâniaBrazil
  2. 2.School of MathematicsThe University of BirminghamBirminghamUK

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