Geodesic
The subdifferential and the directional derivatives of the maximum of a~family of convex functions
Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 807-832

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The paper deals with calculating the directional derivatives and the subdifferential of the maximum of a family of convex functions under weakened compactness conditions on the indexing set (or in the absence of such conditions). 
@article{IM2_1998_62_4_a6,
     author = {V. N. Solov'ev},
     title = {The subdifferential and the directional derivatives of the maximum of a~family of convex functions},
     journal = {Izvestiya. Mathematics },
     pages = {807--832},
     publisher = {mathdoc},
     volume = {62},
     number = {4},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1998_62_4_a6/}
}
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V. N. Solov'ev. The subdifferential and the directional derivatives of the maximum of a~family of convex functions. Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 807-832. http://geodesic.mathdoc.fr/item/IM2_1998_62_4_a6/