Skip to main content
SpringerLink
Log in

Fejér-Type Iterative Processes in the Constrained Quadratic Minimization Problem

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The paper presents an overview of methods for solving an ill-posed problem of constrained convex quadratic minimization based on the Fejér-type iterative methods, which widely use the ideas and approaches developed in the works of I. I. Eremin, the founder of the Ural research school of mathematical programming. Along with a problem statement of general form, we consider variants of the original problem with constraints in the form of systems of equalities and inequalities, which have numerous applications. In addition, particular formulations of the problem are investigated, including the problem of finding a metric projection and solving a linear program, which are of independent interest. A distinctive feature of these methods is that not only convergence but also stability with respect to errors in the input data are established for them; i.e., the methods generate regularizing algorithms in contrast to the direct methods, which do not have this property.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. V. V. Vasin, Foundations of the Theory of Ill-Posed Problems (Izd. SO RAN, Novosibirsk, 2020) [in Russian].

    Google Scholar 

  2. V. V. Vasin and A. L. Ageev, Ill-Posed Problems with A Priori Information (VSP, Utrecht, 1995).

    Book  Google Scholar 

  3. C. T. Lawson and R. J. Hansen, Solving Least Squares Problem (SIAM, Philadelphia, 1995).

    Book  Google Scholar 

  4. L. Fejér, “Über die Lage der Nullstellen von Polynomen, die aus Minimumforderung gewisser Art entspringen,” Math. Ann. 85 (1), 41–48 (1992). https://doi.org/10.1007/BF01449600

    Article  Google Scholar 

  5. T. S. Motzkin and J. J. Schoenberg, “The relaxation method for linear inequalities,” Canad. J. Math. 6 (3), 393–404 (1954). https://doi.org/10.4153/CJM-1954-038-x

    Article  MathSciNet  Google Scholar 

  6. S. Agmon, “The relaxation method for linear inequality,” Canad. J. Math. 6 (3), 382–392 (1954). https://doi.org/10.4153/CJM-1954-037-2

    Article  MathSciNet  Google Scholar 

  7. I. I. Eremin, “Generalization of the relaxation method of Motzkin and Agmon,” Usp. Mat. Nauk 20 (2), 183–187 (1965).

    MathSciNet  Google Scholar 

  8. I. I. Eremin, “Methods of Fejér’s approximations in convex programming,” Math. Notes 3 (2), 139–149 (1968). https://doi.org/10.1007/BF01094336

    Article  Google Scholar 

  9. I. I. Eremin, Theory of Linear Optimization (Izd. Ekaterinburg, Yekaterinburg, 1999; VSP, Utrecht, 2002).

    Google Scholar 

  10. I. I. Eremin, Systems of Linear Inequalities and Linear Optimization (Izd. UrO RAN, Yekaterinburg, 2007) [in Russian].

    Google Scholar 

  11. V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type: Theory and Applications (De Gruyter, Berlin, 2009).

    Book  Google Scholar 

  12. V. V. Vasin, “Iterative methods for solving ill-posed problems with a priori information in Hilbert spaces,” USSR Comput. Math. Math. Phys. 28 (4), 6–13 (1988).

    Article  MathSciNet  Google Scholar 

  13. B. Eicke, Konvex-restringierte schlechtgestellte Probleme und ihre Regularizierung durch Iterationverfahren, Doctoral Dissertation (Berlin, 1991).

  14. B. Martinet, “Determination approachee d’un point fixe d’une application pseudo-contractante. Cas de l’application prox,” C. R. Acad. Sci. Paris, Ser. A–B 274, A163–A165 (1972).

    Google Scholar 

  15. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Amer. Math. Soc. 73 (4), 591–597 (1967). https://doi.org/10.1090/S0002-9904-1967-11761-0

    Article  MathSciNet  Google Scholar 

  16. B. Halperin, “Fixed points of nonexpansive maps,” Bull. Amer. Math. Soc. 73 (6), 957–961 (1967). https://doi.org/10.1090/S0002-9904-1967-11864-0

    Article  MathSciNet  Google Scholar 

  17. G. M. Vainikko, “Error estimates for the method of successive approximation for ill-posed problems,” Autom. Remote Control, 41 (3), 356–363 (1980).

    Google Scholar 

  18. V. G. Karmanov, Mathematical Programming (Nauka, Fizmatlit, 2008) [in Russian].

    Google Scholar 

  19. A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

Download references

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 620108, Yekaterinburg, Russia

    V. V. Vasin

Authors
  1. V. V. Vasin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Vasin.

Ethics declarations

The author of this work declares that he has no conflicts of interest.

Additional information

Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 29, No. 3, pp. 26 - 41, 2023 https://doi.org/10.21538/0134-4889-2023-29-3-26-41.

Translated by M. Deikalova

Publisher's Note Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vasin, V.V. Fejér-Type Iterative Processes in the Constrained Quadratic Minimization Problem. Proc. Steklov Inst. Math. 323 (Suppl 1), S305–S320 (2023). https://doi.org/10.1134/S008154382306024X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154382306024X

Keywords

Search

Navigation