On using exterior penalty approaches for solving linear programming problems

In this paper, we investigate three exterior penalty function approaches for solving linear programming problems. These methods are an active set ℓ2 penalty approach (ASL2), an inequality–equality-based ℓ2 penalty approach (IEL2), and an augmented Lagrangian approach (ALAG). Particular effective variants are explored for each method, along with comments and experience on alternative algorithmic strategies that were empirically investigated. Our motivation is that heretofore, there has been no empirical evidence on how variants of these exterior point algorithms that have been suitably tuned might perform with respect to solving moderately large-scale linear programming problems. Our experiments using a set of randomly generated problems of varying sizes and density, as well as a set of NETLIB test problems, provide insights into the relative performance of these different approaches derived from the basic ℓ2 penalty function, and into their viability for solving linear programs. Average rank tests based on the computational results obtained are performed using two different statistics which assess the speed of convergence and the quality or accuracy of the solution. Overall, a particular variant (ALAG2) of ALAG yielded the best performance with respect to the joint criteria of speed of convergence and the quality of the final solution attained. On the other hand, IEL2 invariably achieved the best quality solutions using comparatively designed termination criteria. However, for highly dense linear programming problems, ASL2 appeared to be the best-alternative among the tested algorithms. Moreover, although our implementation was rudimentary in comparison with CPLEX 6.0, a commercial solver, the tested methods were competitive (sometimes favorable) in attaining a final (though only near optimal) solution for the set of large-scale low-density problems.