zbMATH — the first resource for mathematics

Measure of quasistability of a vector integer linear programming problem with generalized principle of optimality in the Helder metric. (English) Zbl 1214.90087
Summary: A vector integer linear programming problem is considered, the principle of optimality of which is defined by a partitioning of partial criteria into groups with Pareto preference relation within each group and the lexicographic preference relation between them. Quasistability of the problem is investigated. This type of stability is a discrete analog of Hausdorff lower semicontinuity of the many-valued mapping that defines the choice function. A formula of quasistability radius is derived for the case of metric \(l_p\), \(1\leq p\leq\infty\), defined in the space of parameters of the vector criterion. Similar formulae have been obtained before only for combinatorial (Boolean) problems with various kinds of parametrization of the principles of optimality in the cases of \(l_1\) and \(l_\infty\) metrics [S. E. Bukhtoyarov, V. A. Emelichev and Yu. V. Stepanishina, Cybern. Syst. Anal. 39, No. 4, 604–614 (2003); translation from Kibern. Sist. Anal. 39, No. 4, 155–166 (2003; Zbl 1090.90170); S. E. Bukhtoyarov and V. A. Emelichev, Russ. Math. 48, No. 1, 23–27 (2004); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2004, No. 1, 25–30 (2004; Zbl 1096.90538); V. A. Emelichev and K. G. Kuzmin, “Measure of quasistability in \(l_1\) metric vector combinatorial problem with parametric optimality principle”, Izv. Vuzov, Mathematica 12, 3–10 (2005); Cybern. Syst. Anal. 42, No. 4, 609–614 (2006); translation from Kibern. Sist. Anal. 42, No. 4, 175–181 (2006; Zbl 1119.90029)], and for some game theory problems [V. A. Emelichev and A. A. Platonov, Rev. Anal. Numér. Théor. Approx. 35, No. 2, 131–139 (2006; Zbl 1174.90751); A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Moskau: Verlag “Nauka” (1972; Zbl 0235.46001); S. E. Bukhtoyarov and V. A. Emelichev, “Optimality principle (“from Pareto to Slator”) parametrization and stability of suitable mapping trajectorial problems”, Discrete Anal. Oper. Res. 10, No. 4, 3–18 (2003)].

90C10 Integer programming
90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization