Cybernetics and Systems Analysis

, Volume 41, Issue 4, pp 551–558 | Cite as

Stability of Vector Problems of Integer Optimization: Relationship with the Stability of Sets of Optimal and Nonoptimal Solutions

  • T. T. Lebedeva
  • N. V. Semenova
  • T. I. Sergienko
Systems Analysis


Several types of stability against perturbations of vector criterion coefficients are analyzed from the same point of view for a vector integer optimization problem with quadratic criterion functions. The concept of stability is defined. Necessary and sufficient conditions are formulated and analyzed for each type of stability. The topological structure of the sets of initial data on which some solution remains optimal is described.


multicriterion integer optimization partial quadratic criterion functions stability by vector-valued criterion perturbations of initial data 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Stability and Parametric Analysis of Discrete Optimization Problems [in Russian], Naukova Dumka, Kiev (1995).Google Scholar
  2. 2.
    L. N. Kozeratskaya, “Set of strictly efficient points of mixed integer vector optimization problem as a measure of problem's stability,” Cyb. Syst. Anal., Vol. 33, No.6, 901–904 (1997).zbMATHMathSciNetGoogle Scholar
  3. 3.
    V. A. Emelichev and Yu. V. Nikulin, “Stability kernel of the quadratic vector problem of Boolean programming,” Cyb. Syst. Anal., Vol. 37, No.2, 214–219 (2001).MathSciNetGoogle Scholar
  4. 4.
    V. A. Emelichev, E. Girlich, Yu. V. Nikulin, and D. V. Podkopaev, “Stability and regularization of vector problems of integer linear programming,” Optimization, 51, No.4, 645–676 (2002).CrossRefMathSciNetGoogle Scholar
  5. 5.
    T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, “Criterion stability of vector problems of integer quadratic programming,” in: Theory of Optimal Solutions: A Collection of Scientific Papers [in Russian], Kiev, V. M. Glushkov Inst. Cybern. NASU, No. 2 (2003), pp. 140–146.Google Scholar
  6. 6.
    T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, “Optimality and solvability criteria in problems of linear optimization with a convex admissible set,” Dop. NANU, No. 10, 80–85 (2003).Google Scholar
  7. 7.
    T. T. Lebedeva and T. I. Sergienko, “Comparative analysis of different types of stability with respect to constraints of a vector integer-optimization problem,” Cyb. Syst. Anal., Vol. 40, No. 1, 52–57 (2004).MathSciNetGoogle Scholar
  8. 8.
    T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, “Stability of vector integer optimization problems with quadratic criterion functions,” Theory of Stochastic Processes, 10(26), No. 34, 95–101 (2004).Google Scholar
  9. 9.
    V. V. Podinovskii and V. D. Nogin, Pareto-Optimal Solutions of Multicriteria Problems [in Russian], Nauka, Moscow (1982).Google Scholar
  10. 10.
    S. Smale, “Global analysis and economics, V. Pareto theory with constraints,” J. Math. Econ., No. 1, 213–221 (1974).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • T. T. Lebedeva
    • 1
  • N. V. Semenova
    • 1
  • T. I. Sergienko
    • 1
  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKievUkraine

Personalised recommendations