Journal of Optimization Theory and Applications

, Volume 162, Issue 2, pp 392–404 | Cite as

A Subgradient-Like Algorithm for Solving Vector Convex Inequalities

  • J. Y. Bello Cruz
  • L. R. Lucambio Pérez


In this paper, we propose a strongly convergent variant of Robinson’s subgradient algorithm for solving a system of vector convex inequalities in Hilbert spaces. The advantage of the proposed method is that it converges strongly, when the problem has solutions, under mild assumptions. The proposed algorithm also has the following desirable property: the sequence converges to the solution of the problem, which lies closest to the starting point and remains entirely in the intersection of three balls with radius less than the initial distance to the solution set.


Projection methods Strong convergence Subgradient algorithm Vector convex functions 



The authors were partially supported by Project PROCAD-nf-UFG/UnB/IMPA, by Project PRONEX-CNPq-FAPERJ and by Project CAPES-MES-CUBA 226/2012 “Modelos de Otimização e Aplicações”.

The authors would like to extend their gratitude toward anonymous referees whose suggestions helped us to improve the presentation of this paper.


  1. 1.
    Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969) CrossRefGoogle Scholar
  2. 2.
    von Neumann, J.: Functional Operators. The Geometry of Orthogonal Spaces, vol. 2. Princeton University Press, Princeton (1950) Google Scholar
  3. 3.
    Censor, Y., Herman, G.T.: Block-iterative algorithms with underrelaxed Bregman projections. SIAM J. Optim. 13, 283–297 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Robinson, S.M.: A subgradient algorithm for solving K-convex inequalities. In: Optimization and Operations Research. Lecture Notes in Economics and Mathematical Systems, vol. 117, pp. 237–245. Springer, Berlin (1976) CrossRefGoogle Scholar
  6. 6.
    Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent method for nonsmooth convex minimization in Hilbert spaces. Numer. Funct. Anal. Optim. 32, 1009–1018 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent direct method for monotone variational inequalities in Hilbert spaces. Numer. Funct. Anal. Optim. 30, 23–36 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Bolintinéanu, S.: Approximate efficiency and scalar stationarity in unbounded nonsmooth convex vector optimization problems. J. Optim. Theory Appl. 106, 265–296 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Graña Drummond, L.M., Maculan, N., Svaiter, B.F.: On the choice of parameters for the weighting method in vector optimization. Math. Program. 111, 201–216 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Jahn, J.: Scalarization in vector optimization. Math. Program. 29, 203–218 (1984) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989) Google Scholar
  13. 13.
    Luc, D.T.: Scalarization of vector optimization problems. J. Optim. Theory Appl. 55, 85–102 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Bello Cruz, J.Y., Lucambio Pérez, L.R., Melo, J.G.: Convergence of the projected gradient method for quasiconvex multiobjective optimization. Nonlinear Anal. 74, 5268–5273 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Allgower, E.L., Böhmer, K., Potra, F.-A., Rheinboldt, W.C.: A mesh-independence principle for operator equations and their discretizations. SIAM J. Numer. Anal. 23, 160–169 (2011) CrossRefGoogle Scholar
  16. 16.
    Allgower, E.L., Böhmer, K.: Application of the mesh-independence principle to mesh refinement strategies. SIAM J. Numer. Anal. 24, 1335–1351 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Laumen, M.: Newton’s mesh independence principle for a class of optimal shape design problems. SIAM J. Control Optim. 37, 1070–1088 (1987) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Henry, J., Yvon, J.-P.: System Modelling and Optimization. Lecture Notes in Control and Information Sciences, vol. 197. Springer, London (1994) CrossRefzbMATHGoogle Scholar
  19. 19.
    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gordon, R., Herman, G.T.: Reconstruction of pictures from their projections. Commun. ACM 14, 759–768 (1971) CrossRefzbMATHGoogle Scholar
  21. 21.
    Hudson, H.M., Larkin, R.S.: Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans. Med. Imaging 13, 601–609 (1994) CrossRefGoogle Scholar
  22. 22.
    Rockmore, A.J., Macovski, A.: A maximum likelihood approach to transmission image reconstruction from projections. IEEE Trans. Nucl. Sci. 24, 1929–1935 (1977) CrossRefGoogle Scholar
  23. 23.
    Luc, D.T., Tan, N.X., Tinh, P.N.: Convex vector functions and their subdifferential. Acta Math. Vietnam. 23, 107–127 (1998) zbMATHMathSciNetGoogle Scholar
  24. 24.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011) zbMATHGoogle Scholar
  25. 25.
    Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (2007) zbMATHGoogle Scholar
  26. 26.
    Eichfelder, G., Jahn, J.: Vector optimization problems and their solution concepts. In: Recent Developments in Vector Optimization, vol. 1, pp. 1–27. Springer, Berlin (2012) CrossRefGoogle Scholar
  27. 27.
    Chen, G.-Y., Craven, B.D.: A vector variational inequality and optimization over an efficient set. ZOR. Math. Methods Oper. Res. 34, 1–12 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Alber, Ya.I., Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–37 (1998) zbMATHMathSciNetGoogle Scholar
  29. 29.
    Bello Cruz, J.Y., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46, 247–263 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2007) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Universidade Federal de GoiásGoiâniaBrazil
  2. 2.Instituto de Matemática e EstatísticaUniversidade Federal de Goiás, Campus SamambaiaGoiâniaBrazil

Personalised recommendations