A class of convergent primal-dual subgradient algorithms for decomposable convex programs

In this paper we develop a primal-dual subgradient algorithm for preferably decomposable, generally nondifferentiable, convex programming problems, under usual regularity conditions. The algorithm employs a Lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties, in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal-dual optimal solutions. Several classical types of penalty functions are shown to satisfy these specified properties. A geometric convergence rate is established for the algorithm under some additional assumptions. This approach has three principal advantages. Firstly, both primal and dual solutions are available which prove to be useful in several contexts. Secondly, the choice of step sizes, which plays an important role in subgradient optimization, is guided more determinably in this method via primal and dual information. Thirdly, typical subgradient algorithms suffer from the lack of an appropriate stopping criterion, and so the quality of the solution obtained after a finite number of steps is usually unknown. In contrast, by using the primal-dual gap, the proposed algorithm possesses a natural stopping criterion.