The space-fixed genetic algorithm originally proposed by Niesse and Mayne [J. Chem. Phys. 105, 4700 (1996)] is modified and used to study the lowest energy structure of small silicon clusters by employing empirical interatomic potentials. In this new space-fixed genetic algorithm, a gradient-free simplex method, rather than the conventional gradient-driven conjugate gradient minimization employed by Niesse and Mayne, is selected by virtue of its flexibility and applicability to any form of interatomic potentials for which the calculation of derivatives is difficult. Using two empirical three-body potentials, we calculated the ground state structure up to Si-15 successfully using this new genetic algorithm based on the simplex method. The effect of angular dependent three-body potentials on the cluster structures is examined and compared with the experimental results.