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.