This article presents a novel heuristic for constrained optimization of computationally expensive random simulation models. One output is selected as objective to be minimized, while other outputs must satisfy given threshold values. Moreover, the simulation inputs must be integer and satisfy linear or nonlinear constraints. The heuristic combines (i) sequentialized experimental designs to specify the simulation input combinations, (ii) Kriging (or Gaussian process or spatial correlation modeling) to analyze the global simulation input/output data resulting from these designs, and (iii) integer nonlinear programming to estimate the optimal solution from the Kriging metamodels. The heuristic is applied to an (s,S) inventory system and a call-center simulation, and compared with the popular commercial heuristic OptQuest embedded in the Arena versions 11 and 12. In these two applications the novel heuristic outperforms OptQuest in terms of number of simulated input combinations and quality of the estimated optimum.