Summary: In this paper, we give two kinds of sequential optimality theorems for a convex optimization problem, which are expressed in terms of sequences of \(\varepsilon\)-subgradients and subgradients of involved functions. The involved functions of the problem are proper, lower-semicontinuous and convex. We give sufficient conditions for the closedness of characterization cones for the problem. Moreover, we characterize the solution set for the problem. Several numerical examples are presented to illustrate results.