Geodesic
A criterion for checking if a convex set belongs to the union of a finite number of convex sets
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1571-1578 
D. G. Pivovarchuk. A criterion for checking if a convex set belongs to the union of a finite number of convex sets. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1571-1578. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A necessary and sufficient condition for the inclusion of a convex compact set in the union of a finite number of convex sets is proved. This condition obtained using the convex analysis techniques is a condition on the maximin of a given function. Using the dynamic programming, checking this condition is reduced to evaluating a set of functions and checking a condition for their values. The reduced form of the criterion is more convenient from the computational point of view.

[1] Pontryagin L. S., “Lineinye differentsialnye igry presledovaniya”, Matem. sb., 112:3 (1980), 307–330 | MR | Zbl

[2] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977 | MR | Zbl

[3] Pivovarchuk D. G., “Upravlenie lineinoi sistemoi pri neopredelennosti s nefiksirovannym vremenem popadaniya na tselevoe mnozhestvo”, Differents. ur-niya, 42:12 (2006), 1735–1744 | MR | Zbl

[4] Rokafellar R., Vypuklyi analiz, Mir, M., 1973

[5] Vasilev F. P., Metody optimizatsii, Faktorial, M., 2002

[6] Bellman R., Dinamicheskoe programmirovanie, Izd-vo inostr. lit., M., 1960 | MR

[7] Gabasov R., Kirillova F. M., Osnovy dinamicheskogo programmirovaniya, Izd-vo Belorusskogo un-ta, Minsk, 1975 | MR | Zbl