This paper studies integrated random search algorithms for continuous optimization-via-simulation (COvS) problems. We first tailor the Gaussian process-based search (GPS) algorithm to handle COvS problems. We then analyze the potential sampling issue of the GPS algorithm and propose to construct a desirable Gaussian mixture model (GMM) which is amenable for efficient sampling and at the same time also maintains the desirable property of exploitation and exploration trade-off. Then, we propose a Gaussian mixture model-based random search (GMRS) algorithm. We build global convergence of both the tailored GPS algorithm and the GMRS algorithm for COvS problems. Finally, we carry out some numerical studies to illustrate the performance of the GMRS algorithm.