In this paper, I first characterize the nonemptiness and boundedness of the weakly efficient solution set of convex vector optimization problem, especially a cone-constrained convex vector optimization problem in infinite-dimensional real reflexive Banach spaces. More specifically, I will consider the covex vector optimization problem when the objective space R_m is ordered by a nontrivial, polyhedral cone with nonempty interior instead of the nonnegative orthant R₊m. Then, I apply the characterizations to the convergence analysis of a class of penalty methods.