Restricted Normal Cones and the Method of Alternating Projections: Theory


In this paper, we introduce and develop the theory of restricted normal cones which generalize the classical Mordukhovich normal cone. We thoroughly study these objects from the viewpoint of constraint qualifications and regularity. Numerous examples are provided to illustrate the theory. This work provides the theoretical underpinning for a subsequent article in which these tools are applied to obtain a convergence analysis of the method of alternating projections for nonconvex sets.

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  1. 1.

    Bauschke, H.H., Borwein, J.M., Lewis, A.S.: The method of cyclic projections for closed convex sets in Hilbert space. In: Censor, Y., Reich, S. (eds.) Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem 1995), Contemporary Mathematics vol. 204, pp. 1–38. American Mathematical Society (1997)

  2. 2.

    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)

  3. 3.

    Bauschke, H.H., Luke, D.R., Phan, H.M., Wang, X.: Restricted normal cones and sparsity optimization with affine constraints. Found. Comput. Math. (2012, in press). arXiv preprint

  4. 4.

    Bauschke, H.H., Luke, D.R., Phan, H.M., Wang, X.: Restricted Normal Cones and the Method of Alternating Projections: Applications. preprint (2013). doi:10.1007/s11228-013-0238-3

  5. 5.

    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer-Verlag (2005)

  6. 6.

    Censor, Y., Zenios, S.A.: Parallel Optimization. Oxford University Press (1997)

  7. 7.

    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer-Verlag (1998)

  8. 8.

    Deutsch, F.: The angle between subspaces of a Hilbert space. In: Approximation Theory, Wavelets and Applications (Maratea, 1994). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences vol. 454, pp. 107–130. Kluwer (1995)

  9. 9.

    Deutsch, F.: Best Approximation in Inner Product Spaces. Springer (2001)

  10. 10.

    Dixmier, J.: Étude sur les variétés et les opérateurs de Julia, avec quelques applications. Bull. Soc. Math. Fr. 77, 11–101 (1949)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Friedrichs, K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Amer. Math. Soc. 41, 321–364 (1937)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9, 485–513 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Lewis, A.S., Malick, J.: Alternating projection on manifolds. Mathematics of Operations Research 33, 216–234 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Loewen, P.D.: Optimal Control via Nonsmooth Analysis. CRM Proceedings & Lecture Notes, AMS, Providence, RI (1993)

    Google Scholar 

  15. 15.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer-Verlag (2006)

  16. 16.

    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  17. 17.

    Rockafellar, R.T.,Wets, R.J-B.: Variational Analysis. Springer, corrected 3rd printing (2009)

  18. 18.

    von Neumann, J.: Functional Operators Vol. II. The Geometry of Orthogonal Spaces. Annals of Mathematical Studies #22, Princeton University Press, Princeton (1950)

    Google Scholar 

  19. 19.

    Wiener, N.: On the factorization of matrices. Commentarii Mathematici Helvetici 29, 97–111 (1955)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing (2002)

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Correspondence to Heinz H. Bauschke.

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Bauschke, H.H., Luke, D.R., Phan, H.M. et al. Restricted Normal Cones and the Method of Alternating Projections: Theory. Set-Valued Var. Anal 21, 431–473 (2013).

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  • Constraint qualification
  • Convex set
  • Friedrichs angle
  • Normal cone
  • Nonconvex set
  • Projection operator
  • Restricted normal cone
  • Superregularity

Mathematics Subject Classifications (2010)

  • Primary 49J52; Secondary 47H09
  • 90C26