Generalized moment problems –called feature moments in the area of machine learning– are here considered with and without relaxation. The solution is the minimum of \(\varphi \) -divergences, according to an extended Maximum Entropy Principle. Inference from sampled data constraints leads to balance the divergence by a relaxation term. In the literature, the form of the minimizing solution is obtained by resorting to Fenchel’s duality theorem or the method of Lagrange multipliers. An alternative method, resorting to Hilbert spaces, presented here, yields a necessary and sufficient condition under second order assumptions. It is based on a decomposition via a nested procedure of the relaxed problem into two successive problems, one of which without relaxation.