  We developed a corporative stochastic approximation (CSA) type algorithm for
semi-infinite programming (SIP), where the cut generation problem is solved
inexactly. First, we provide general error bounds for inexact CSA. Then, we
propose two specific random constraint sampling schemes to approximately solve
the cut generation problem. When the objective and constraint functions are
generally convex, we show that our randomized CSA algorithms achieve an
$\mathcal{O}(1/\sqrt{N})$ rate of convergence in expectation (in terms of
optimality gap as well as SIP constraint violation). When the objective and
constraint functions are all strongly convex, this rate can be improved to
$\mathcal{O}(1/N)$.
