Journal of Global Optimization

, Volume 52, Issue 4, pp 831–842 | Cite as

Generalized projections onto convex sets

  • O. P. Ferreira
  • S. Z. Németh


This paper introduces the notion of projection onto a closed convex set associated with a convex function. Several properties of the usual projection are extended to this setting. In particular, a generalization of Moreau’s decomposition theorem about projecting onto closed convex cones is given. Several examples of distances and the corresponding generalized projections associated to particular convex functions are presented.


Projection Convex set Convex cone Convex function 


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  1. 1.
    Arioli M., Duff I., Noailles J., Ruiz D.: A block projection method for sparse matrices. SIAM J. Sci. Stat. Comput. 13(1), 47–70 (1992)CrossRefGoogle Scholar
  2. 2.
    Bauschke H.H., Borwein J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)CrossRefGoogle Scholar
  3. 3.
    Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming, Theory and Algorithms. Wiley, New York (1979)Google Scholar
  4. 4.
    Berk R., Marcus R.: Dual cones, dual norms, and simultaneous inference for partially ordered means. J. Am. Stat. Assoc. 91(433), 318–328 (1996)CrossRefGoogle Scholar
  5. 5.
    Butnariu D., Kassay G.: A proximal-projection method for finding zeros of set-valued operators. SIAM J. Control Optim. 47(4), 2096–2136 (2008)CrossRefGoogle Scholar
  6. 6.
    Carrizosa E., Plastria F.: Optimal expected-distance separating halfspace. Math. Oper. Res. 33(3), 662–677 (2008)CrossRefGoogle Scholar
  7. 7.
    Censor Y., Elfving T.: Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem. SIAM J. Matrix Anal. Appl. 24(1), 40–58 (2002) (electronic)CrossRefGoogle Scholar
  8. 8.
    Censor, Y., Elfving, T., Herman, G.T., Nikazad, T.: On diagonally relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30(1), 473–504 (2007/08)Google Scholar
  9. 9.
    Censor Y., Gordon D., Gordon R.: Component averaging: an efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27(6), 777–808 (2001)CrossRefGoogle Scholar
  10. 10.
    Csiszár I.: Generalized projections for non-negative functions. Acta Math. Hung. 68(1-2), 161–186 (1995)CrossRefGoogle Scholar
  11. 11.
    Dax A.: The distance between two convex sets. Linear Algebra Appl. 416(1), 184–213 (2006)CrossRefGoogle Scholar
  12. 12.
    Dax A., Sreedharan V.P.: Theorems of the alternative and duality. J. Optim. Theory Appl. 94(3), 561–590 (1997)CrossRefGoogle Scholar
  13. 13.
    Dykstra R.L.: An algorithm for restricted least squares regression. J. Am. Stat. Assoc. 78(384), 837–842 (1983)CrossRefGoogle Scholar
  14. 14.
    Foley J.D., van Dam A., Feiner S.K., Hughes J.F.: Computer Graphics: Principles and Practice. Addison-Wesley systems programming series, Reading (1990)Google Scholar
  15. 15.
    Hiriart-Urruty J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms: Fundamentals. I, volume 305 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1990)Google Scholar
  16. 16.
    Isac G., Németh A.B.: Monotonicity of metric projections onto positive cones of ordered Euclidean spaces. Arch. Math. Basel 46(6), 568–576 (1986)CrossRefGoogle Scholar
  17. 17.
    Isac G., Németh A.B.: Isotone projection cones in Euclidean spaces. Ann. Sci. Math. Québec 16(1), 35–52 (1992)Google Scholar
  18. 18.
    Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and their Applications, volume 88 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980)Google Scholar
  19. 19.
    Mangasarian O.L.: Arbitrary-norm separating plane. Oper. Res. Lett. 24(1–2), 15–23 (1999)CrossRefGoogle Scholar
  20. 20.
    Mangasarian O.L.: Polyhedral boundary projection. SIAM J. Optim. 9(4), 1128–1134 (1999) (electronic, Dedicated to John E. Dennis, Jr., on his 60th birthday)CrossRefGoogle Scholar
  21. 21.
    Moreau J.-J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C R Acad. Sci. Paris 255, 238–240 (1962)Google Scholar
  22. 22.
    Németh A.B., Németh S.Z.: How to project onto an isotone projection cone. Linear Algebra Appl. 433(1), 41–51 (2010)CrossRefGoogle Scholar
  23. 23.
    Németh S.Z.: Iterative methods for nonlinear complementarity problems on isotone projection cones. J. Math. Anal. Appl. 350(1), 340–347 (2009)CrossRefGoogle Scholar
  24. 24.
    Németh S.Z.: Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum. Acta Math. Hung. 127(4), 376–390 (2010)CrossRefGoogle Scholar
  25. 25.
    Németh S.Z.: Isotone retraction cones in Hilbert spaces. Nonlinear Anal. 73(2), 495–499 (2010)CrossRefGoogle Scholar
  26. 26.
    Pardalos, P.M., Rassias, T.M, Khan, A.A (eds): Nonlinear Analysis and Variational Problems, volume 35 of Springer Optimization and its Applications. Springer, New York (2010) (In honor of George Isac)Google Scholar
  27. 27.
    Plastria F., Carrizosa E.: Gauge distances and median hyperplanes. J. Optim. Theory Appl. 110(1), 173–182 (2001)CrossRefGoogle Scholar
  28. 28.
    Rami M.A., Helmke U., Moore J.B.: A finite steps algorithm for solving convex feasibility problems. J. Global Optim. 38(1), 143–160 (2007)CrossRefGoogle Scholar
  29. 29.
    Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ (1970)Google Scholar
  30. 30.
    Scolnik H.D., Echebest N., Guardarucci M.T., Vacchino M.C.: Incomplete oblique projections for solving large inconsistent linear systems. Math. Program 111(1-2, Ser. B), 273–300 (2008)CrossRefGoogle Scholar
  31. 31.
    Stewart G.W.: On the perturbation of pseudo-inverses, projections and linear least squares problems. SIAM Rev. 19(4), 634–662 (1977)CrossRefGoogle Scholar
  32. 32.
    Tao Y., Liu G.-P., Chen W.: Globally optimal solutions of max-min systems. J. Global Optim. 39(3), 347–363 (2007)CrossRefGoogle Scholar
  33. 33.
    Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets. In: Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), pp. 237–341. Academic Press, New York (1971)Google Scholar

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© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.IME/UFGGoiâniaBrazil
  2. 2.School of MathematicsThe University of BirminghamEdgbaston, BirminghamUnited Kingdom

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