  Solving convex Semi-Infinite Programming (SIP) problems is challenging when
the separation problem, i.e., the problem of finding the most violated
constraint, is computationally hard. We propose to tackle this difficulty by
solving the separation problem approximately, i.e., by using an inexact oracle.
Our focus lies in two algorithms for SIP, namely the Cutting-Planes (CP) and
the Inner-Outer Approximation (IOA) algorithms. We prove the CP convergence
rate to be in O(1/k), where k is the number of calls to the limited-accuracy
oracle, if the objective function is strongly convex. Compared to the CP
algorithm, the advantage of the IOA algorithm is the feasibility of its
iterates. In the case of a semi-infinite program with Quadratically Constrained
Quadratic Programming separation problem, we prove the convergence of the IOA
algorithm toward an optimal solution of the SIP problem despite the oracle's
inexactness.
