In the paper it is considered a convex programming problem (CPP) with functional and non-functional constraints. In contrast to previous works, in the study of convex optimization problems, we do not deal with the classical approach of perturbations. In particular, thanks to the new representation of the indicator function on a convex set, the successful use of the infimal convolution method in this work plays a key role in proving duality results for problem CPP. Also, we consider a convex mathematical programming problem with inequality and linear equality constraints given by some matrix. In this case, it turns out that the dual cone to the cone of tangent directions coincides with the set of the image of the points of transposed matrix, taken with a minus sign.