Taiwanese Journal of Mathematics

THE SUBGRADIENT METHOD FOR SOLVING VARIATIONAL INEQUALITIES WITH COMPUTATIONAL ERRORS IN A HILBERT SPACE

Alexander J. Zaslavski

Full-text: Open access

Abstract

In a Hilbert space, we study the asymptotic behavior of the subgradient method for solving a variational inequality, under the presence of computational errors. Most results known in the literature establish convergence of optimization algorithms, when computational errors are summable. In the present paper, the convergence of the subgradient method to the solution of a variational inequalities is established for nonsummable computational errors. We show that the the subgradient method generates good approximate solutions, if the sequence of computational errors is bounded from above by a constant.

Article information

Source
Taiwanese J. Math., Volume 16, Number 5 (2012), 1781-1790.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406796

Digital Object Identifier
doi:10.11650/twjm/1500406796

Mathematical Reviews number (MathSciNet)
MR2970684

Zentralblatt MATH identifier
1255.58006

Subjects
Primary: 58E35: Variational inequalities (global problems) 65K15: Numerical methods for variational inequalities and related problems

Keywords
computational error Hilbert space subgradient method variational inequality

Citation

Zaslavski, Alexander J. THE SUBGRADIENT METHOD FOR SOLVING VARIATIONAL INEQUALITIES WITH COMPUTATIONAL ERRORS IN A HILBERT SPACE. Taiwanese J. Math. 16 (2012), no. 5, 1781--1790. doi:10.11650/twjm/1500406796. https://projecteuclid.org/euclid.twjm/1500406796


Export citation

References

  • \item[1.] Ya. I. Alber, A. N. Iusem and M. V. Solodov, On the projected subgradient method for nonsmooth convex optimization in a Hilbert space, Math. Programming, 81 (1998), 23-35.
  • \item[2.] Ya. Alber and I. Ryazantseva, Nonlinear ill-posed problems of monotone type, Springer, 2006.
  • \item[3.] K. Barty, J.-S. Roy and C. Strugarek, Hilbert-valued perturbed subgradient algorithms, Math. Oper. Res., 32 (2007), 551-562.
  • \item[4.] R. Burachik, L. M. Grana Drummond, A. N. Iusem and B. F. Svaiter, Full convergence of the steepest descent method with inexact line searches, Optimization, 32 (1995), 137-146.
  • \item[5.] R. S. Burachik, J. O. Lopes, G. J. P Da Silva, An inexact interior point proximal method for the variational inequality, Comput. Appl. Math., 28 (2009), 15-36.
  • \item[6.] L. C. Ceng, B. S. Mordukhovich and J. C. Yao, Hybrid approximate proximal method with auxiliary variational inequality for vector optimization, J. Optim. Theory Appl., 146 (2010), 267-303.
  • \item[7.] Y. Censor and A. Gibali, Projections onto super-half-spaces for monotone variational inequality problems in finite-dimensional spaces, Journal of Nonlinear and Convex Analysis, 9, (2008), 461-475.
  • \item[8.] V. F. Demyanov and L. V. Vasil'ev, Nondifferentiable optimization, Nauka, Moscow 1981.
  • \item[9.] F. Facchinei and J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, volume I and volume II, Springer-Verlag, New York, 2003,
  • \item[10.] M. Fukushima, A relaxed projection method for variational inequalities, Mathematical Programming, 35 (1986), 58-70.
  • \item[11.] E. Huebner and R. Tichatschke, Relaxed proximal point algorithms for variational inequalities with multi-valued operators, Optim. Methods Softw., 23 (2008), 847-877.
  • \item[12.] A. N. Iusem and E. Resmerita, A proximal point method in nonreflexive Banach spaces, Set-Valued Var. Anal., 18 (2010), 109-120.
  • \item[13.] A. Kaplan and R. Tichatschke, Bregman-like functions and proximal methods for variational problems with nonlinear constraints, Optimization, 56 (2007), 253-265.
  • \item[14.] K. C. Kiwiel, Convergence of approximate and incremental subgradient methods for convex optimization, SIAM J. Optim., 14, 807-840.
  • \item[15.] P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.
  • \item[16.] B. T. Polayk, Introduction to optimization Optimization Software, New York, 1987.
  • \item[17.] A. J. Zaslavski, The projected subgradient method for nonsmooth convex optimization in the presence of computational errors, Numerical Functional Analysis and Optimization, 31 (2010), 616-633.
  • \item[18.] A. J. Zaslavski, Convergence of a proximal method in the presence of computational errors in Hilbert spaces SIAM J. Optimization, 20 (2010), 2413-2421.