Convex separable minimization subject to bounded variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint "less than or equal to" and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality "greater than or equal to" constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.