In this paper, we consider a type of cone-constrained convex program in finite-dimensional space, and are interested in characterization of the solution set of this convex program with the help of the Lagrange multiplier. We establish necessary conditions for a feasible point being an optimal solution. Moreover, some necessary conditions and sufficient conditions are established which simplifies the corresponding results in Jeyakumar et al. (J Optim Theory Appl 123(1), 83103, 2004). In particular, when the cone reduces to three specific cones, that is, the p -order cone, p cone and circular cone, we show that the obtained results can be achieved by easier ways by exploiting the special structure of those three cones.