Simulation optimization refers to the iterative procedure in search of the optimal parameter when the objective function can only be evaluated by stochastic simulation. STRONG (Stochastic Trust Region Response Surface Convergent Method) is a newly developed design-of-experiments based simulation optimization method. It incorporates the idea of trust region method (TRM) for deterministic optimization into the traditional response surface methodology (RSM) to eliminate the human intervention required by RSM and to achieve the desired convergence. In the earlier paper, we proved the convergence of STRONG and demonstrated its computational efficiency. The original STRONG assumes the stochastic response follows a normal distribution. This paper relaxes the normal assumption and develops a framework called STRONG-X which is applicable for generally distributed additive noise with bounded variance. The convergence of STRONG-X can be proved and the generality of STRONG-X makes it an appealing method.