Applied Mathematics & Optimization

, Volume 78, Issue 3, pp 613–641 | Cite as

Linear Regularity and Linear Convergence of Projection-Based Methods for Solving Convex Feasibility Problems

  • Xiaopeng Zhao
  • Kung Fu Ng
  • Chong LiEmail author
  • Jen-Chih Yao


For a finite/infinite family of closed convex sets with nonempty intersection in Hilbert space, we consider the (bounded) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem. Several sufficient conditions are provided to ensure the bounded linear regularity in terms of the interior-point conditions and some finite codimension assumptions. A unified projection method, called Algorithm B-EMOPP, for solving the convex feasibility problem is proposed, and by using the bounded linear regularity, the linear convergence results for this method are established under a new control strategy introduced here.


Convex feasibility problem Linear regularity Projection algorithm 

Mathematics Subject Classification

47H09 65J05 41A25 90C25 



The authors are grateful to the anonymous referee for his valuable suggestions and remarks, especially for providing the relevant references [8, 9, 26], which helped to improve the presentation of the paper. The Xiaopeng Zhao was supported in part by the National Natural Science Foundation of China (Grant 11626168). The Kung Fu Ng was supported in part by an Earmarked Grant (GRF) from the Research Grant Council of Hong Kong (Project Nos. CUHK 14304014 and 14302516), and CUHK direct Grant 4053218. The Chong Li was supported in part by the National Natural Science Foundation of China (Grant 11571308). The Jen-Chih Yao was in part supported by the National Science Council of Taiwan under Grant MOST 105-2115-M-039-002-MY3.


  1. 1.
    Agmon, S.: The relaxation method for linear inequalities. Can. J. Math. 6, 382–392 (1954)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amemiya, I., Ando, T.: Convergence of random products of contractions in Hilbert space. Acta Sci Math (Szeged) 26, 239–244 (1965)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Auslender, A. M\(\acute{e}\)thodes Num\(\acute{e}\)riques pour la R\(\acute{e}\)solution des Probl\(\grave{e}\)mes d’Optimisation avec Contraintes, Th\(\grave{e}\)se, Facult\(\acute{e}\) des Sciences, Grenoble (1969)Google Scholar
  4. 4.
    Bakan, A., Deutsch, F., Li, W.: Strong CHIP, normality and linear regularity of convex sets. Trans. Am. Math. Soc. 357, 3831–3863 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bauschke, H.H.: Projection algorithms and monotone operators. Ph. D. thesis, Department of Mathematics, Simon Fraser University, Burnaby, BC (1996).
  6. 6.
    Bauschke, H.H.: A norm convergence result on random products of relaxed projections in Hilbert space. Trans. Am. Math. Soc. 347, 1365–1373 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program Ser. A 86, 135–160 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bauschke, H.H., Borwein, J.M., Tseng, P.: Bounded linear regularity, strong CHIP, and CHIP are distinct properties. J. Convex Anal. 7, 395–412 (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bauschke, H.H., Combettes, P.L., Kruk, S.G.: Extrapolation algorithm for affine-convex feasibility problems. Numer. Algorithms 41, 239–274 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Borwein, J.M., Li, G.Y., Yao, L.J.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets. SIAM J. Optim. 24, 498–527 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688–692 (1965)zbMATHGoogle Scholar
  14. 14.
    Chui, C., Deutsch, F., Ward, J.D.: Constrained best approximation in Hilbert space. Constr. Approx. 6, 35–64 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chui, C., Deutsch, F., Ward, J.D.: Constrained best approximation in Hilbert space II. J. Approx. Theory 71, 231–238 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  17. 17.
    Combettes, P.L.: The foundations of set theoretic estimation. Proc. IEEE 81, 182–208 (1993)CrossRefGoogle Scholar
  18. 18.
    Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 95, pp. 155–270. Academic Press, New York (1996)Google Scholar
  19. 19.
    Combettes, P.L.: Hilbertian convex feasibility problem: convergence of projection methods. Appl. Math. Optim. 35, 311–330 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Deutsch, F.: The method of alternating orthogonal projections. In: Singh, S.P. (ed.) Approximation Theory, Spline Functions and Application, vol. 356, pp. 105–121. Kluwer, The Netherlands (1992)CrossRefGoogle Scholar
  21. 21.
    Deutsch, F., Li, W., Ward, J.D.: A dual approach to constrained interpolation from a convex subset of Hilbert space. J. Approx. Theory 90, 385–414 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7, 1–24 (1967)CrossRefGoogle Scholar
  23. 23.
    Hiriart-Urruty, J., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Grundlehren der Math Wiss, vol. 305. Springer-Verlag, New York (1993)zbMATHGoogle Scholar
  24. 24.
    Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49, 263–265 (1952)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hu, Y.H., Li, C., Yang, X.Q.: On convergence rates of linearized proximal algorithms for convex composite optimization with applications. SIAM J. Optim. 26, 1207–1235 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kiwiel, K., Lopuch, B.: Surrogate projection methods for finding fixed points of firmly nonexpansive mappings. SIAM J. Optim. 7, 1084–1102 (1997)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li, C., Ng, K.F., Pong, T.K.: The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 18, 643–665 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li, C., Ng, K.F.: Strong CHIP for infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 16, 311–340 (2005)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Li, C., Ng, K.F.: The dual normal CHIP and linear regularity for infinite systems of convex sets in Banach spaces. SIAM J. Optim. 24, 1075–1101 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Motzkin, T.S., Schoenberg, I.J.: The relaxation method for linear inequalities. Can. J. Math. 6, 393–404 (1954)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ng, K.F., Yang, W.H.: Regularities and their relations to error bounds. Math. Program Ser. A 99, 521–538 (2004)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ottavy, N.: Strong convergence of projection-like methods in Hilbert spaces. J. Optim. Theory Appl. 56, 433–461 (1988)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Rockafellar, R.T.: Convex Analysis. Priceton University Press, Priceton, NJ (1970)CrossRefGoogle Scholar
  34. 34.
    von Neumann, J.: On rings of operators. Reduction theory. Ann. Math. 50:401–485. (1949) (the result of interest first appeared in 1933 in Lecture Notes)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wang, J.H., Hu, Y.H., Li, C., Yao, J.C.: Linear convergence of CQ algorithms and applications in gene regulatory network inference. Inverse Probl. 33, 055017 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Yosida, K.: Functional Analysis. Grundlehren der Math. Wiss, vol. 123. Springer, New York (1980)zbMATHGoogle Scholar
  37. 37.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge, NJ (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Xiaopeng Zhao
    • 1
  • Kung Fu Ng
    • 2
  • Chong Li
    • 3
    Email author
  • Jen-Chih Yao
    • 4
    • 5
  1. 1.Department of MathematicsTianjin Polytechnic UniversityTianjinPeople’s Republic of China
  2. 2.Department of MathematicsChinese University of Hong KongHong KongPeople’s Republic of China
  3. 3.School of Mathematical SciencesZhejiang UniversityHangzhouPeople’s Republic of China
  4. 4.Center for General EducationChina Medical UniversityTaichungTaiwan
  5. 5.Research Center for Nonlinear Analysis and OptimizationKaohsiung Medical UniversityKaohsiungTaiwan

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