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Stability radius of an efficient solution of a vector problem of integer linear programming in the Hölder metric. (English. Russian original) Zbl 1119.90029
Cybern. Syst. Anal. 42, No. 4, 609-614 (2006); translation from Kibern. Sist. Anal. 42, No. 4, 175-181 (2006).
Summary: A vector (multicriterion) problem of integer linear programming is considered on a finite set of feasible solutions. A metric \(l_p,\;1\leq p\leq \infty\) , is defined on the parameter space of the problem. A formula of the maximum permissible level of perturbations is obtained for the parameters that preserve the efficiency (Pareto optimality) of a given solution. Necessary and sufficient conditions of two types of stability of the problem are obtained as corollaries.

MSC:
90C10 Integer programming
90C31 Sensitivity, stability, parametric optimization
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[1] I. V. Sergienko, Mathematical Models and Methods of Solution of Discrete Optimization Problems [in Russian], Naukova Dumka, Kiev (1988).
[2] I. V. Sergienko, L. N. Kozeratskaya, and T. T. Lebedeva, Investigation of Stability and Parametrical Analysis of Discrete Optimization Problems [in Russian], Naukova Dumka, Kiev (1995). · Zbl 0864.90096
[3] I. V. Sergienko, Informatics in Ukraine: Formation, Development, and Problems [in Ukrainian], Naukova Dumka, Kyiv (1999).
[4] I. V. Sergienko and V. P. Shilo, Discrete Optimization Problems: Issues, Solution Methods, and Investigations [in Russian], Naukova Dumka, Kiev (2003).
[5] T. T. Lebedeva and T. I. Sergienko, ”Comparative analysis of different types of stability with respect to constraints of a vector integer-optimization problem,” Kibern. Sist. Anal., No. 1, 63–70 (2004). · Zbl 1066.90118
[6] T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, ”Stability of vector integer optimization problems with quadratic criterion functions,” Theory of Stochastic Processes, 3, No. 4, 95–101, Inst. of Mathematics of NASU, Kyiv (2004). · Zbl 1076.90034
[7] T. T. Lebedeva, N. V. Semenova, and T. I. Sergienko, ”Stability of vector integer-optimization problems: Interrelation with the stability of optimal and nonoptimal solution sets,” Kibern. Sist. Anal., No. 4, 90–100 (2005). · Zbl 1098.90066
[8] V. A. Emelichev and Yu. V. Nikulin, ”Stability of an efficient solution of a vector problem of integer linear programming,” Dokl. NAN Belarusi, 44, No. 4, 26–28 (2000). · Zbl 1177.90295
[9] V. A. Emelichev, K. G. Kuzmin, and A. M. Leonovich, ”Stability in vector combinatorial optimization problems,” Avtomat. Telemekh., No. 2, 79–92 (2004).
[10] V. A. Emelichev, V. N. Krichko, and Yu. V. Nikulin, ”The stability radius of an efficient solution in minimax Boolean programming problem,” Control and Cybernetics, 33, No. 1, 127–132 (2004). · Zbl 1115.90035
[11] V. A. Emelichev and K. G. Kuzmin, ”Stability radius of an efficient solution of a vector problem of Boolean programming in the metric l 1,” Dokl. Ross. Akad. Nauk, 401, No. 6, 733–735 (2005).
[12] V. A. Emelichev, K. G. Kuzmin, and Yu. V. Nikulin, ”Stability analysis of the Pareto optimal solution for some vector Boolean optimization problem,” Optimization, 54, No. 6, 545–561 (2005). · Zbl 1147.90378
[13] V. A. Emelichev and K. G. Kuzmin, ”Stability radius of a lexicographic optimum of a vector problem of Boolean programming,” Kibern. Sist. Anal., No. 2, 71–81 (2005). · Zbl 1076.90033
[14] V. A. Emelichev and K. G. Kuzmin, ”Sensitivity analysis of an efficient solution to a vector Boolean problem of minimization of projections of linear functions on R + and R ” Diskr. Anal. i Issled. Oper., Ser. 2, 12, No. 2, 24–43 (2005). · Zbl 1249.90165
[15] V. A. Emelichev and K. G. Kuzmin, ”Finite coalition games with a parametric conception of equilibrium under uncertainty,” Izv. Ross. Akad. Nauk, Ser. Teor. i Sist. Upr., No. 2, 96–101 (2006).
[16] L. N. Kozeratskaya, T. T. Lebedeva, and T. I. Sergienko, ”Problems of integer programming with a vector criterion: Parametrical analysis and investigation of stability,” Dokl. Akad. Nauk SSSR, 307, No. 3, 527–529 (1989).
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