In reliability engineering studies, computer codes are increasingly used to model physical phenomena which, in many cases, can be very time-consuming to run. A widely accepted approach consists in approximating the CPU-time expensive computer model by a surrogate model. One of the most popular surrogate model is the Gaussian Process regression, as it provides, additionally to a prediction at an unobserved point, an uncertainty around this prediction (a predictive distribution). However, in practice, the quality of this metamodel depends on several choices, as the estimation and validation algorithms. The present work aims at proposing a new algorithm, based on constrained optimization multi-objective techniques, to estimate the Gaussian process hyperparameters in order to ensure robust and reliable predictive distribution of the Gaussian process. An intensive numerical benchmark on various analytical functions, with different input dimensions and learning sample sizes, shows its good performance in comparison with standard estimation algorithms. The new algorithm is also applied to a real test case modeling an aquatic ecosystem. It is compared with a recent robust and sophisticated Bayesian method; it proves to be as efficient while being less sensitive to the specification of the Gaussian process model.