In this paper, several Lagrange multiplier characterizations of the solution set of a convex infinite programming problem are given. Characterizations of solution sets of cone-constrained convex programs are derived as well. The procedure is then adopted to a semi-convex problem with convex constraints. For this problem, we present firstly a necessary and sufficient condition for optimality and secondly a characterization of its optimal solution set, based on a Lagrange multiplier associated with a given solution and on directional derivatives of the objective function.