Many problems arising from communication system design can be formulated as optimization problems. In practice, one is often interested in not only the numerical solution to the problems but also the special structure of their optimal solution. In this talk, we shall use some examples from wireless communications and information theory to show that exploring the Lagrangian dual of these (convex) problems often reveal the structure of their optimal solution and the structure of the optimal solution will further lead to better algorithms for solving the corresponding problems.