Projection onto simplicial cones by a semi-smooth Newton method
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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.
KeywordsMetric 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.
- 9.Clarke, F.H.: Optimization and Nonsmooth Analysis. Classics in applied mathematics, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)Google Scholar
- 10.Dennis, J.E. Jr., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations, volume 16 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1996) (Corrected reprint of the 1983 original)Google Scholar
- 13.Ekárt, A., Németh, A.B., Németh, S.Z.: Rapid heuristic projection on simplicial cones (2010) arXiv:1001.1928
- 14.Foley, J.D., van Dam, A., Feiner, S.K., Hughes, J.F.: Computer Graphics: Principles and Practice. Addison-Wesley systems programming series (1990)Google Scholar
- 16.Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms: Fundamentals. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], volume 305, Springer, Berlin (1993)Google Scholar
- 27.Murty, K.G.: Sigma Series in Applied Mathematics. Linear complementarity, linear and nonlinear programming. Heldermann, Berlin (1988)Google Scholar
- 35.Ujvári, M.: On the projection onto a finitely generated cone, 2007, Preprint WP 2007–5. MTA SZTAKI, Laboratory of Operations Research and Decision Systems, Budapest (2007)Google Scholar