Subdifferentials of Convex Operators Valued in the Space of Integrable Functions with Applications to Risk-Averse Optimization

We study differentiability properties of convex operators defined on a Banach space with values in an L_(p) space and of their compositions with monotonic convex functionals on this space. We develop new tools for operators enjoying an additional feature known as the local property. The new approach and results go beyond the classical theory of normal integrands and lattice-valued operators. We further describe the subdifferentials of compositions of such operators with convex monotonic functionals. The new results are applied to obtain novel optimality conditions in the subdifferential form for a broad class of risk-averse stochastic optimization problems with risk functionals as objectives, with partial information, and with stochastic dominance constraints. While our analysis is motivated by the theory and methods of risk-averse optimization, it addresses problems of a more general structure and has a potential for further applications