Abstract
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.
Similar content being viewed by others
References
Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969)
von Neumann, J.: Functional Operators. The Geometry of Orthogonal Spaces, vol. 2. Princeton University Press, Princeton (1950)
Censor, Y., Herman, G.T.: Block-iterative algorithms with underrelaxed Bregman projections. SIAM J. Optim. 13, 283–297 (2002)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
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)
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)
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)
Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)
Bolintinéanu, S.: Approximate efficiency and scalar stationarity in unbounded nonsmooth convex vector optimization problems. J. Optim. Theory Appl. 106, 265–296 (2000)
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)
Jahn, J.: Scalarization in vector optimization. Math. Program. 29, 203–218 (1984)
Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)
Luc, D.T.: Scalarization of vector optimization problems. J. Optim. Theory Appl. 55, 85–102 (1987)
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)
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)
Allgower, E.L., Böhmer, K.: Application of the mesh-independence principle to mesh refinement strategies. SIAM J. Numer. Anal. 24, 1335–1351 (1987)
Laumen, M.: Newton’s mesh independence principle for a class of optimal shape design problems. SIAM J. Control Optim. 37, 1070–1088 (1987)
Henry, J., Yvon, J.-P.: System Modelling and Optimization. Lecture Notes in Control and Information Sciences, vol. 197. Springer, London (1994)
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)
Gordon, R., Herman, G.T.: Reconstruction of pictures from their projections. Commun. ACM 14, 759–768 (1971)
Hudson, H.M., Larkin, R.S.: Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans. Med. Imaging 13, 601–609 (1994)
Rockmore, A.J., Macovski, A.: A maximum likelihood approach to transmission image reconstruction from projections. IEEE Trans. Nucl. Sci. 24, 1929–1935 (1977)
Luc, D.T., Tan, N.X., Tinh, P.N.: Convex vector functions and their subdifferential. Acta Math. Vietnam. 23, 107–127 (1998)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)
Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (2007)
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)
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)
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)
Bello Cruz, J.Y., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46, 247–263 (2010)
Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2007)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bello Cruz, J.Y., Lucambio Pérez, L.R. A Subgradient-Like Algorithm for Solving Vector Convex Inequalities. J Optim Theory Appl 162, 392–404 (2014). https://doi.org/10.1007/s10957-013-0300-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0300-1