Enhancing Lagrangian Dual Optimization for Linear Programs by Obviating Nondifferentiability

We consider nondifferentiable optimization problems that arise when solving Lagrangian duals of large–scale linear programs. Different from traditional subgradient–based approaches, we design two new methods that attempt to circumvent or obviate the nondifferentiability of the objective function, so that standard differentiable optimization techniques could be used. These methods, called the perturbation technique and the barrier–Lagrangian reformulation, are implemented as initialization procedures to provide a warm start to a theoretically convergent nondifferentiable optimization algorithm. Our computational study reveals that this two–phase strategy produces much better solutions with less computation in comparison with both the stand–alone nondifferentiable optimization procedure employed, and the popular Held–Wolfe–Crowder subgradient heuristic. Furthermore, the best version of this composite algorithm is shown to consume only about 3.19% of the CPU time required by the commercial linear programming solver CPLEX 8.1 (using the dual simplex option) to produce the same quality solutions. We also demonstrate that this initialization technique greatly facilitates quick convergence in the primal space when used as a warm start for ergodic–type primal recovery schemes.