This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a
sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.