We consider the convex optimization problem when the objective function may not smooth and the constraint set is represented by constraint functions that are locally Lipschitz and directionally differentiable, but neither necessarily concave nor continuously differentiable. The obtained results improve and extend those results that have been presented in [Dutta, J., Lalitha, C.S.:Optimality conditions in convex optimization revisited. Optim. Lett. 7(2),221-229 (2013)], and [Dutta, J.:Barrier method in nonsmooth convex optimization without convex representation. Optim. Lett. 9(6), 1177-1185 (2015)], by removing the regularity and continuously differentiable assumptions on the constraint functions from the considering.