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

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