A Penalty Based Simplex Method for Linear Programming.
We give a general description of a new advanced implementation of the simplex method for linear programming. The method ``decouples'' a notion of the simplex basic solution into two independent entities: a solution and a basis. This generalization makes it possible to incorporate new strategies into the algorithm since the iterates no longer need to be the vertices of the simplex. An advantage of such approach is a possibility of taking steps along directions that are not simplex edges (in principle they can even cross the interior of the feasible set). It is exploited in our new approach to finding the initial solution in which global infeasibility is handled through a dynamically adjusted penalty term. We present several new techniques that have been incorporated into the method. These features include: previously mentioned method for finding an initial solution, an original approximate steepest edge pricing algorithm, dynamic adjustment of the penalty term. The presence of the new crashing and restart procedures based on the penalty term make the algorithm particularly suitable for sequential ``warm start'' calls when solving subproblems in decomposition approaches. The same features may be used in post optimal analysis. The efficiency of the new features is demonstrated when running the method on a subset of difficult linear programs from the NETLIB collection.
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