A Simplex-Genetic method for solving the Klee-Minty cube

Although the Simplex Method (SM) developed for Dantzig is efficient for solving many linear programming problems (LPs), there are constructions of hard linear programs as the Klee-Minty cubes and another deformed products, where this method has an exponential behavior. This work presents the integration of genetic algorithms (GA) and SM to fastly reach the optimum of this type of problems. This integration, called Simplex Genetic Method (SGM), applies first a GA to find a solution near the optimum and afterwards uses the SM to reach the optimum in a few steps. In the GA phase, the populations are constructed by only basic LP solutions codified as binary chromosomes and the crossover operator uses a tabu approach over infeasible solutions to produce the new offsprings. Based in this binary representation, a translation schema is used to transfer the GA solution as the initial solution of the Simplex search mechanism, avoiding that the SM realizes many iterations and reducing the optimum search time. In this work, several instances of the Klee-Minty cube are evaluated and compared with the traditional SM and the results suggest that for hard linear problems the SGM has better behavior that the SM. Key-Words: Genetic algorithms, linear programming, Simplex method, optimization.