  We develop two adaptive discretization algorithms for convex semi-infinite
optimization, which terminate after finitely many iterations at approximate
solutions of arbitrary precision. In particular, they terminate at a feasible
point of the considered optimization problem. Compared to the existing finitely
feasible algorithms for general semi-infinite optimization problems, our
algorithms work with considerably smaller discretizations and are thus
computationally favorable. Also, our algorithms terminate at approximate
solutions of arbitrary precision, while for general semi-infinite optimization
problems the best possible approximate-solution precision can be arbitrarily
bad. All occurring finite optimization subproblems in our algorithms have to be
solved only approximately, and continuity is the only regularity assumption on
our objective and constraint functions. Applications to parametric and
non-parametric regression problems under shape constraints are discussed.
