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

TL;DR

This paper investigates the existence of Lagrange multipliers in infinite-dimensional optimization problems under Gâteaux differentiability, providing new conditions based on calmness and Gâteaux derivatives.

Contribution

It introduces novel sufficient conditions for Lagrange multiplier existence using Gâteaux derivatives and calmness assumptions in infinite-dimensional settings.

Findings

Established existence of Lagrange multipliers under calmness conditions

Derived sufficient conditions using Gâteaux derivatives

Applied results to optimization and stochastic control problems

Abstract

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.