The main goal of this study is to find the most effective set of parameters for the Simplified Generalized Simulated Annealing algorithm, SGSA, when applied to distinct cost function as well as to find a possible correlation between the values of these parameters sets and some topological characteristics of the hypersurface of the respective cost function. The SGSA algorithm is an extended and simplified derivative of the GSA algorithm, a Markovian stochastic process based on Tsallis statistics that has been used in many classes of problems, in particular, in biological molecular systems optimization. In all but one of the studied cost functions, the global minimum was found in 100% of the 50 runs. For these functions the best visiting parameter, qV, belongs to the interval [1.2, 1.7]. Also, the temperature decaying parameter, qT, should be increased when better precision is required. Moreover, the similarity in the locus of optimal parameter sets observed in some functions indicates that possibly one could extract topological information about the cost functions from these sets.