The authors consider the convex infinite optimization problem
\[ \operatorname{Min} f(x) \;\text{ s.t }\;f_t(x) \leq 0,\;\forall t\in T \text{ and }x \in C, \tag{P} \]
where \(X\) is a locally convex Hausdorff topological vector space, \(f\), \(f_t: X \rightarrow\mathbb R \cup (+\infty)\), \(t \in T\), are proper, lower semi-continuous and convex functions, \(C\) is a nonempty, closed convex subset of \(X\), and \(T\) is an arbitrary index set. Several characterizations of the solution set of (P) are given together with the characterization of the solution set of (P) when a minimizing sequence (instead of a solution of (P)) is known. These results are also obtained for cone-constrained convex programs. They also obtain optimality condition for a semi-convex program with convex constraints and give characterizations of its solution set.