Abstract
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.
Similar content being viewed by others
REFERENCES
I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Stability and Parametric Analysis of Discrete Optimization Problems [in Russian], Naukova Dumka, Kiev (1995).
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).
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).
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).
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.
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).
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).
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).
V. V. Podinovskii and V. D. Nogin, Pareto-Optimal Solutions of Multicriteria Problems [in Russian], Nauka, Moscow (1982).
S. Smale, “Global analysis and economics, V. Pareto theory with constraints,” J. Math. Econ., No. 1, 213–221 (1974).
Author information
Authors and Affiliations
Additional information
The study was supported from the State Fund for Basic Research of Ukraine (Grant F7/275-2001).
__________
Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 90–100, July–August 2005.
Rights and permissions
About this article
Cite this article
Lebedeva, T.T., Semenova, N.V. & Sergienko, T.I. Stability of Vector Problems of Integer Optimization: Relationship with the Stability of Sets of Optimal and Nonoptimal Solutions. Cybern Syst Anal 41, 551–558 (2005). https://doi.org/10.1007/s10559-005-0090-z
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10559-005-0090-z