In industrial production, one usually wants to seek an optimal product recipe or operation condition; however, due to the possible or known presence of multiple local optima in an unknown system such as a newly developed fermentation process, one may need to find the best global solution via the global optimization approach. An effective-global optimizer of non-convex functions can be applied to an unknown system with constraints to reach the global optimum by obtaining the surrogate model experimentally. Nevertheless, large experiments are usually indispensable for achieving a defined target. In this work, a monitoring chart describing objective function values with respect to cluster centers of local and global minima (or maxima) is proposed to follow the development of the identified radial basis function (RBF) model which is based on the information gathering from experiments specially designed. Based on the monitoring chart, whether the region surrounding the global extreme is reached can be followed. The proposed optimizing algorithm to reach the global optimum in an unknown process consists of the following two steps. Initially, the experiments designed by the global optimizer (rbfSolve routine in TOMLAB/CGO) is conducted before the region surrounding the global extreme is reached. When the region of global extreme is approaching, additional-optimizing experiments designed by the identified RBF model are then carried out to accelerate the rate to achieve the global optimum. The performance of the optimizing algorithm and the monitoring chart on an unknown process with the constraints proposed in this work was evaluated through (a) a constrained multimodal function as a problem of finding the recipe for a newly developed product and (b) a feed-rate optimization of a fed-batch fermentation process as a problem in obtaining an optimal-feeding trajectory. One can conclude that the experimental approach for achieving global optimization of an unknown process with constraints via an RBF based method is achievable in limited experiments.