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Generalized projections onto convex sets

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Abstract

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.

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References

  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)

    Article  Google Scholar 

  2. Bauschke H.H., Borwein J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)

    Article  Google Scholar 

  3. Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming, Theory and Algorithms. Wiley, New York (1979)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  6. Carrizosa E., Plastria F.: Optimal expected-distance separating halfspace. Math. Oper. Res. 33(3), 662–677 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  10. Csiszár I.: Generalized projections for non-negative functions. Acta Math. Hung. 68(1-2), 161–186 (1995)

    Article  Google Scholar 

  11. Dax A.: The distance between two convex sets. Linear Algebra Appl. 416(1), 184–213 (2006)

    Article  Google Scholar 

  12. Dax A., Sreedharan V.P.: Theorems of the alternative and duality. J. Optim. Theory Appl. 94(3), 561–590 (1997)

    Article  Google Scholar 

  13. Dykstra R.L.: An algorithm for restricted least squares regression. J. Am. Stat. Assoc. 78(384), 837–842 (1983)

    Article  Google Scholar 

  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. 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. Isac G., Németh A.B.: Monotonicity of metric projections onto positive cones of ordered Euclidean spaces. Arch. Math. Basel 46(6), 568–576 (1986)

    Article  Google Scholar 

  17. Isac G., Németh A.B.: Isotone projection cones in Euclidean spaces. Ann. Sci. Math. Québec 16(1), 35–52 (1992)

    Google Scholar 

  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. Mangasarian O.L.: Arbitrary-norm separating plane. Oper. Res. Lett. 24(1–2), 15–23 (1999)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Németh A.B., Németh S.Z.: How to project onto an isotone projection cone. Linear Algebra Appl. 433(1), 41–51 (2010)

    Article  Google Scholar 

  23. Németh S.Z.: Iterative methods for nonlinear complementarity problems on isotone projection cones. J. Math. Anal. Appl. 350(1), 340–347 (2009)

    Article  Google Scholar 

  24. Németh S.Z.: Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum. Acta Math. Hung. 127(4), 376–390 (2010)

    Article  Google Scholar 

  25. Németh S.Z.: Isotone retraction cones in Hilbert spaces. Nonlinear Anal. 73(2), 495–499 (2010)

    Article  Google Scholar 

  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. Plastria F., Carrizosa E.: Gauge distances and median hyperplanes. J. Optim. Theory Appl. 110(1), 173–182 (2001)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  29. Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

  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)

    Article  Google Scholar 

  31. Stewart G.W.: On the perturbation of pseudo-inverses, projections and linear least squares problems. SIAM Rev. 19(4), 634–662 (1977)

    Article  Google Scholar 

  32. Tao Y., Liu G.-P., Chen W.: Globally optimal solutions of max-min systems. J. Global Optim. 39(3), 347–363 (2007)

    Article  Google Scholar 

  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)

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Authors and Affiliations

  1. 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, United Kingdom

    S. Z. Németh

Authors
  1. O. P. Ferreira
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  2. S. Z. Németh
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Correspondence to O. P. Ferreira.

Additional information

This work was done while the first author was visiting the second author at the School of Mathematics in The University of Birmingham, during the period of January 2010 to July 2010.

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Ferreira, O.P., Németh, S.Z. Generalized projections onto convex sets. J Glob Optim 52, 831–842 (2012). https://doi.org/10.1007/s10898-011-9714-1

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