Metric regularity under G\^ateaux differentiability with applications to optimization and stochastic optimal control problems

The main objective of this work is to study the existence of Lagrange multipliers for infinite dimensional problems under G^ateux differentiability assumptions on the data. Our investigation follows two main steps: the proof of the existence of Lagrange multipliers under a calmness assumption on the constraints and the study of sufficient conditions, which only use the G^ateaux derivative of the function defining the constraint, that ensure this assumption.