Geodesic
Optimization problems with averaging over the variables
Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 2, pp. 227-238

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The problems of nonlinear programming, criteria and limitations depend on the variables averaged. It is shown that if these problems have solutions, the Lagrangian reaches the maximum for the variables, which are averaged. The functions defining the problem can not be differentiable and continuous on these variables, the set of possible values may contain isolated points. In variational problems there can be no solution in the class of piecewise continuous functions of the variables, but there can be a generalized solution in which these variables change in the sliding mode, and the optimality criterion tends to its upper edge. If in such problems the solution in the class of piecewise-continuous functions exists, the conditions of optimality of this solution are in the form of the Hamiltonian function of the maximum principle. The relationship between the average over time and across multiple variables is considered.
Keywords: the average optimization, expansion of the set of admissible equivalence extension, variation of probability measures, the conditions in the form of the maximum principle. 
@article{MAIS_2017_24_2_a7,
     author = {A. M. Tsirlin},
     title = {Optimization problems with averaging over the variables},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {227--238},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2017_24_2_a7/}
}
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A. M. Tsirlin. Optimization problems with averaging over the variables. Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 2, pp. 227-238. http://geodesic.mathdoc.fr/item/MAIS_2017_24_2_a7/