A minimization problem with a convex separable objective function subject to linear equality constraints and box constraints (bounds on the variables) is considered. Necessary and sufficient optimality condition is proved for a feasible solution to be an optimal solution to this problem. Primal-dual analysis is also included. Examples of some convex separable objective functions for the considered problem are presented. 
On the solution of multidimensional convex separable continuous knapsack problem with bounded variables