WIAS Preprint No. 2755, (2020)

Optimality conditions for convex stochastic optimization problems in Banach spaces with almost sure state constraint


  • Geiersbach, Caroline
  • Wollner, Winnifried

2010 Mathematics Subject Classification

  • 49K20 49K21 49K45 49N15 49J53


  • PDE-constrained optimization under uncertainty, optimization in Banach spaces, optimality conditions, convex stochastic optimization in Banach spaces, two-stage stochastic optimization, regular Lagrange multipliers, duality




We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty.

Appeared in

Download Documents