Geodesic
Measure of quasistability of a~vector integer linear programming problem with generalized principle of optimality in the Helder metric
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2008), pp. 58-67.

Voir la notice de l'article provenant de la source Math-Net.Ru

A vector integer linear programming problem is considered, 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 had 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 [1–4], and for some game theory problems [5–7]. 
@article{BASM_2008_2_a5,
     author = {Vladimir A. Emelichev and Andrey A. Platonov},
     title = {Measure of quasistability of a~vector integer linear programming problem with generalized principle of optimality in the {Helder} metric},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {58--67},
     publisher = {mathdoc},
     number = {2},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2008_2_a5/}
}
TY  - JOUR
AU  - Vladimir A. Emelichev
AU  - Andrey A. Platonov
TI  - Measure of quasistability of a~vector integer linear programming problem with generalized principle of optimality in the Helder metric
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2008
SP  - 58
EP  - 67
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BASM_2008_2_a5/
LA  - en
ID  - BASM_2008_2_a5
ER  - 
%0 Journal Article
%A Vladimir A. Emelichev
%A Andrey A. Platonov
%T Measure of quasistability of a~vector integer linear programming problem with generalized principle of optimality in the Helder metric
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2008
%P 58-67
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BASM_2008_2_a5/
%G en
%F BASM_2008_2_a5
Vladimir A. Emelichev; Andrey A. Platonov. Measure of quasistability of a~vector integer linear programming problem with generalized principle of optimality in the Helder metric. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2008), pp. 58-67. http://geodesic.mathdoc.fr/item/BASM_2008_2_a5/

[1] Bukhtoyarov S.E., Emelichev V.A., “Stability of discrete vector problems with the parametric principle of optimality”, Cybernetics and Systems Analysis, 39:4 (2003), 604–614 | DOI | MR | Zbl

[2] Bukhtoyarov S.E., Emelichev V.A., “Quasistability of vector trajectory problem with the parametric optimality principle”, Izv. Vuzov, Mathematica, 2004, no. 1, 25–30 | MR | Zbl

[3] Emelichev V.A., Kuzmin K.G., “Measure of quasistability in $l_1$ metric vector combinatorial problem with parametric optimality principle”, Izv. Vuzov, Mathematica, 2005, no. 12, 3–10 (in Russian) | MR | Zbl

[4] Emelichev V.A., Kuzmin K.G., “About stability radius of effective solution vector problem of integer linear programming in Gelder metric”, Cybernetics and System Analysis, 2006, no. 4, 175–181 | Zbl

[5] Emelichev V.A., Platonov A.A., “About one discrete analog of Hausdorff semi-continuity of suitable mapping in a vector combinatorial problem with a parametric principle of optimality (“from Slater to lexicographic”)”, Revue d'Analyse Numerique et de Theorie de l'Approximation, Cluj-Napeca, Romania, 35:2 (2006) | MR

[6] Kolmogorov A.N., Fomin S.V., Elements of function theory and functional analysis, Nauka, Moscow, 1972 (in Russian)

[7] Bukhtoyarov S.E., Emelichev V.A., “Optimality principle (“from Pareto to Slator”) parametrization and stability of suitable mapping trajectorial problems”, Discrete analysis and operation research, 10:4 (2003), 3–18 (in Russian) | MR

[8] Bukhtoyarov S.E., Emelichev V.A., “Measure of stability for a finite cooperative game with a parametric optimality principle (“from Pareto to Nash”)”, Comput. Math. and Math. Phys., 46:7 (2006), 1193–1199 | DOI

[9] Bukhtoyarov S.E., Emelichev V.A., “About stability of a finite cooperative game with parametric conception of equilibrium (“from Pareto to Nash”)”, Rapporteur of the Belarusian NAN, 50:1 (2006), 9–11 (in Russian) | MR

[10] Emelichev V.A., Kuzmin K.G., “Finite cooperative games with a parametric concept of equilibrium under uncertainty conditions”, Journal of Computer and Systems Sciences International, 45:2 (2006), 276–281 | DOI | MR

[11] Bukhtoyarov S.E., Emelichev V.A., “On stability of an optimal situation in a finite cooperative game with a parametric concept of equilibrium (“from lexicographic optimality to Nash equilibrium”)”, Computer Science Journal of Moldova, 12:3 (2004), 371–380 | MR | Zbl

[12] Emelichev V.A., Podkopaev D.P., “On a quantitative measure of stability for a vector problem in integer programming”, Comput. Math. and Math. Phys., 38:11 (1998), 1727–1731 | MR | Zbl

[13] Emelichev V.A., Girlich E., Nikulin Yu.V., Podkopaev D.P., “Stability and regularization of vector problems of integer linear programming”, Optimization, 51:4 (2002), 645–676 | DOI | MR | Zbl

[14] Sergienco I.V., Kozeratskaya L.N., Lebedeva T.T., Investigation of stability and parametric analysis of discrete optimization problem, Naukova Dumka, Kiev, 1995 (in Russian)

[15] Sergienko I.V., Shilo V.P., Discrete optimization problems. Problems, decision methods, investigations, Naukova Dumka, Kiev, 2003 (in Russian)

[16] Smale S., “Global analysis and economics, V. Pareto, theory with constraints”, J. Math. Econom., 1:3 (1974), 213–221 | DOI | MR | Zbl