Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Boris S. Mordukhovich

Dept. of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.
and: Peoples’ Friendship University of Russia, Moscow 117198, Russia

boris@math.wayne.edu

Keywords

  • Integral functionals
  • subdifferential mappings
  • Gelfand integral
  • generalized Leibniz formulas
  • saturated measure spaces
  • Lyapunov convexity theorem
  • stochastic dynamic programming
  • optimal value

Mathematics Subject Classification

49J52, 28B05, 28B20; 90C40, 90C46, 90C56

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Suggested citation

B. S. Mordukhovich, N. Sagara. “Subdifferentials of Nonconvex Integral Functionals in Banach Spaces with Applications to Stochastic Dynamic Programming.” Journal of Convex Analysis 25 (2018), No. 2, 643–673. https://doi.org/10.68381/jca25039

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press.