On the Computational Efficiency of Subgradient Methods: A Case Study in Combinatorial Optimization

Subgradient methods (SM) have long been the preferred way to solve large-scale Nondifferentiable Optimization problems, such as those arising from the solution of Lagrangian duals of hard combinatorial optimization problems. Although other methods exist that show a significantly higher convergence rate in some circumstances, SM have certain unique advantages that may make them competitive under the right set of conditions. Besides, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and, in some applications, practical performances. We aim at computationally evaluating a large class of SM for the specific application of the solution of Lagrangian duals of integer programs, in order to assess if and how the main improvements suggested along the years have changed the computational landscape of these approaches. For this we first propose a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked, but computationally important, features like projection and dynamic generation of variables. This gives us a large class of SM, with several algorithmic parameters. We fine-tune and test them on two Lagrangian dual problems with significantly different characteristics, both arising from the same relevant and hard combinatorial optimization problem the-Fixed-Charge Multicommodity Capacitated Network Design problem-in order to assess the potential impact of the characteristics of the function to be minimized on the optimal algorithmic choices. Our results show that SM can be competitive with more sophisticated approaches when the tolerance required for solution is not particularly tight, which is what happens when Lagrangian techniques are used within exact or heuristic approaches for solving the original hard combinatorial optimization problem. However, for this to happen appropriate and nontrivial tuning has to be performed.