Extending the Farkas Lemma Approach to Necessity Conditions to Infinite Programming

Under mild assumptions, the classical Farkas lemma approach to Lagrange multiplier theory is extended to an infinite programming formulation. The main result generalizes the usual first-order necessity conditions to address problems in which the domain of the objective function is Hilbert space and the number of constraints is arbitrary. The result is used to obtain necessity conditions for a well-known problem from the statistical literature on probability density estimation