Skip to main content
SpringerLink
Log in

Lexicographic regularization and duality for improper linear programming problems

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

Abstract

A new approach to the optimal lexicographic correction of improper linear programming problems is proposed. The approach is based on the multistep regularization of the classical Lagrange function with respect to primal and dual variables simultaneously. The regularized function can be used as a basis for generating new duality schemes for problems of this kind. Theorems on the convergence and numerical stability of the method are presented, and an informal interpretation of the obtained generalized solution is given.

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

Similar content being viewed by others

References

  1. I. I. Eremin, “Duality for improper problems of linear and convex programming,” Soviet Math. Dokl. 23, 62–66 (1981).

    MATH  Google Scholar 

  2. I. I. Eremin, Vl. D. Mazurov, and N. N. Astaf’ev, Improper Problems of Linear and Convex Programming (Nauka, Moscow, 1983) [in Russian].

    MATH  Google Scholar 

  3. I. I. Eremin, Duality in Linear Optimization (Izd. UrO RAN, Yekaterinburg, 2001) [in Russian].

    MATH  Google Scholar 

  4. A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1979; Wiley, New York, 1981).

    MATH  Google Scholar 

  5. F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].

    MATH  Google Scholar 

  6. V. D. Skarin, “An approach to the analysis of improper linear programming problems,” USSR Comp. Math. Math. Phys. 26 (2), 73–79 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  7. L. D. Popov, “Linear correction of ill-posed convexly concave minimax problems using the maximin criterion,” USSR Comp. Math. Math. Phys. 26 (5), 30–39 (1986).

    Article  Google Scholar 

  8. V. D. Skarin, “On a regularization method for inconsistent convex programming problems,” Russ. Math. (Iz. VUZ) 39 (12), 78–85 (1995).

    MathSciNet  MATH  Google Scholar 

  9. E. G. Gol’shtein, Duality Theory in Mathematical Programming and Its Applications (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  10. I. I. Eremin, “Problems in sequential programming,” Sib. Math. J. 14 (1), 36–43 (1973).

    Article  MATH  Google Scholar 

  11. V. V. Fedorov, Numerical Maximin Methods (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  12. J. Guddat, “Stability in convex quadratic parametric programming,” Math. Operationsforsch. Statist. 7 (2), 223–245 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  13. W. S. Dorn, “Duality in quadratic programming,” Quart. Appl. Math. 18, 155–162 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. J. Hoffman, “On approximate solutions of systems of linear inequalities,” J. Res. Nat. Bur. Standards 49, 263–265 (1952).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990, Russia

    L. D. Popov & V. D. Skarin

  2. Ural Federal University, Yekaterinburg, 620000, Russia

    L. D. Popov & V. D. Skarin

Authors
  1. L. D. Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. D. Skarin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. D. Popov.

Additional information

Original Russian Text © L.D. Popov, V.D. Skarin, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 3.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Popov, L.D., Skarin, V.D. Lexicographic regularization and duality for improper linear programming problems. Proc. Steklov Inst. Math. 295 (Suppl 1), 131–144 (2016). https://doi.org/10.1134/S0081543816090145

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543816090145

Keywords

Search

Navigation