This paper focuses on the Gaussian Process regression (GPR) of non-linear functions subject to multiple linear constraints, such as boundedness, monotonicity or convexity. It presents an algorithm allowing to optimize, in a concerted way, the statistical moments of the Gaussian process used for the regression, and the position of a reduced number of points where the constraints are required to hold, such that the constraints are verified in the whole input space, with high probability, at a reasonable computational cost. After having presented the theoretical bases and the numerical implementation of this algorithm, this paper illustrates its efficiency though the analysis of several test functions of increasing dimensions.