  Handling an infinite number of inequality constraints in infinite-dimensional
spaces occurs in many fields, from global optimization to optimal transport.
These problems have been tackled individually in several previous articles
through kernel Sum-Of-Squares (kSoS) approximations. We propose here a unified
theorem to prove convergence guarantees for these schemes. Pointwise
inequalities are turned into equalities within a class of nonnegative kSoS
functions. Assuming further that the functions appearing in the problem are
smooth, focusing on pointwise equality constraints enables the use of
scattering inequalities to mitigate the curse of dimensionality in sampling the
constraints. Our approach is illustrated in learning vector fields with side
information, here the invariance of a set.
