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We develop two adaptive discretization algorithms for convex semi-infiniteoptimization, which terminate after finitely many iterations at approximatesolutions of arbitrary precision. In particular, they terminate at a feasiblepoint of the considered optimization problem. Compared to the existing finitelyfeasible algorithms for general semi-infinite optimization problems, ouralgorithms work with considerably smaller discretizations and are thuscomputationally favorable. Also, our algorithms terminate at approximatesolutions of arbitrary precision, while for general semi-infinite optimizationproblems the best possible approximate-solution precision can be arbitrarilybad. All occurring finite optimization subproblems in our algorithms have to besolved only approximately, and continuity is the only regularity assumption onour objective and constraint functions. Applications to parametric andnon-parametric regression problems under shape constraints are discussed.

Approximate solutions of convex semi-infinite optimization problems in finitely many iterations