Geodesic
The subdifferential and the directional derivatives of the maximum of a~family of convex functions.~II
Izvestiya. Mathematics , Tome 65 (2001) no. 1, pp. 99-121

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The paper deals with calculating the directional derivatives and the subdifferential of the maximum of convex functions with no compactness conditions on the indexing set. We apply our results to the problems of minimax theory in which the Lagrange function is not assumed to be concave. We also apply these results to the duality theory of non-convex extremum problems, and strengthen earlier results of Yakubovich, Matveev and the author. We illustrate our results by investigating a problem of optimal design of experiments. 
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     author = {V. N. Solov'ev},
     title = {The subdifferential and the directional derivatives of the maximum of a~family of convex {functions.~II}},
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V. N. Solov'ev. The subdifferential and the directional derivatives of the maximum of a~family of convex functions.~II. Izvestiya. Mathematics , Tome 65 (2001) no. 1, pp. 99-121. http://geodesic.mathdoc.fr/item/IM2_2001_65_1_a6/