The purpose of this paper is to investigate an iterative regularization method of proximal point type for solving ill posed vector convex optimization problems in Hilbert spaces. Applications to the convex feasibility problems and the problem of common fixed points for nonexpansive potential mappings are also given.
Similar content being viewed by others
References
Nguyen Buong, “Regularization for unconstrained vector optimization of convex functionals in Banach spaces,” Zh. Vychisl. Mat. Mat. Fiz., 46, No. 3, 372–378.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1976).
R. T. Rockafellar, “Monotone operators and proximal point algorithm,” SIAM J. Contr. Optim., 14, 877–897 (1976).
O. Güler, “On the convergence of the proximal point algorithm for convex minimization,” SIAM J. Contr. Optim., 29, 403–419 (1991).
H. H. Bauschke, J. V. Burke, F. R. Deutsch, H. S. Hundal, and J. D. Vanderwerff, “A new proximal point iteration that converges weakly but not in norm,” Proc. Amer. Math. Soc., 133, 1829–1835 (2005).
M. V. Solodov and B. F. Svaiter, “Forcing strong convergence of the proximal point iteration in Hilbert space,” Math. Program., 87, 189–202 (2000).
I. P. Ryazantseva, “Proximal regularization algorithm for nonlinear equations of monotone type,” Zh. Vychisl. Mat. Mat. Fiz., 42, No. 9, 1295–1303 (2002).
F. Alvarez and H. Attouch, “An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping,” Set-Valued Anal., 9, 3–11 (2001).
A. Moudafi, “Second order differential proximal methods for equilibrium problems,” J. Inequalit. Pure Appl. Math., 4, No. 1 (2003).
A. Moudafi and E. Elisabeth, “Approximate inertial proximal methods using the enlargement of maximal monotone operators,” Int. J. Pure Appl. Math., 5, 283–299 (2003).
H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems, ” SIAM Rev., 38, 367–426 (1996).
H. H. Bauschke and S. G. Kruk, “Reflection-projection method for convex feasibility problems with an obtuse cone,” J. Optim. Theory Appl., 120, No. 3, 503–531 (2004).
P. L. Combettes, “Hilbertian convex feasibility problem: convergence of projection methods,” Appl. Math. Optim., 35, 311–330 (1997).
F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets for nonexpansive mappings,” Numer. Funct. Anal. Optim., 19, 33–56 (1998).
H. K. Xu, “An iterative approach to quadratic optimization,” J. Optim. Theory Appl., 116, No. 3, 659–678 (2003).
J. G. O'Hara, P. Pillay, and H. K. Xu, “Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces,” Nonlin. Analysis, 54, No. 8, 1417–1426 (2003).
W. Takahashi, T. Tamura, and M. Toyoda, “Approximation of common fixed points of a family of finite nonexpansive mappings in Banach spaces,” Sci. Math. Jpn., 56, No. 3, 475–480 (2002).
Jong Soo Jung, Yeol Je Cho, and R. P. Agarwal, “Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach spaces,” Fixed Point Theory Appl., 2, 125–135 (2005).
C.E. Chidume, H. Zegeye, and N. Shahzad, “Convergence theorems for a common fixed point of a finite family of non-self nonexpansive mappings,” Fixed Point Theory and Appl., 2, 233–241 (2005).
Author information
Authors and Affiliations
Additional information
Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1275–1281, September, 2008.
Rights and permissions
About this article
Cite this article
Buong, N. Inertial proximal point regularization algorithm for unconstrained vector convex optimization problems. Ukr Math J 60, 1483–1491 (2008). https://doi.org/10.1007/s11253-009-0137-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0137-9