Journal of Convex Analysis 25 (2018), No. 2, 643--673
Copyright Heldermann Verlag 2018
Subdifferentials of Nonconvex Integral Functionals in Banach Spaces with Applications to Stochastic Dynamic Programming
Boris S. Mordukhovich
Dept. of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.
boris@math.wayne.edu
Nobusumi Sagara
Dept. of Economics, Hosei University, Tokyo 194-0298, Japan
nsagara@hosei.ac.jp
The paper concerns the investigation of nonconvex and nondifferentiable integral functionals on general Banach spaces, which may not be reflexive and/or separable. Considering two major subdifferentials of variational analysis, we derive nonsmooth versions of the Leibniz rule on subdifferentiation under the integral sign, where the integral of the subdifferential set-valued mappings generated by Lipschitzian integrands is understood in the Gelfand sense. Besides examining integration over complete measure spaces and also over those with nonatomic measures, our special attention is drawn to a stronger version of measure nonatomicity, known as saturation, to invoke the recent results of the Lyapunov convexity theorem type for the Gelfand integral of the subdifferential mappings. The main results are applied to the subdifferential study of the optimal value functions and deriving the corresponding necessary optimality conditions in nonconvex problems of stochastic dynamic programming with discrete time on the infinite horizon.
Keywords: Integral functionals, subdifferential mappings, Gelfand integral, generalized Leibniz formulas, saturated measure spaces, Lyapunov convexity theorem, stochastic dynamic programming, optimal value.
MSC: 49J52, 28B05, 28B20; 90C40, 90C46, 90C56
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