Abstract: In his talk, I will present a class of problems from stochastic optimization, where the constraints take the form of a family of partial differential equations (PDEs). These problems have a wide range of applications, from engineering, materials science, social sciences, to economics. One central challenge is the analysis in infinite dimensions; optimization theory must delicately balance topologies, and if the PDEs are discretized, optimization procedures should take into account numerical error. Incorporating uncertainty into such problems creates new challenges. I will discuss recent work on optimality theory for convex problems with additional nonsmoothness in the form of a state constraint. I will present optimality conditions and regularization techniques, and show how this analysis can be helpful in the construction of algorithms. Numerical experiments using a path-following stochastic gradient method will be presented, and related work and perspectives will be mentioned.