We consider a multivariable regression model under shape constraints (monotonicity, convexity, positivity,...) built as a linear combination of product of functions of a single variable. For each variable, the functions form a Chebyshev system. We develop an iterative procedure, where at each step the initial shape requirement is approximated by a set of linear constraints. The main result of this paper is that this procedure is shown to converge to the optimal solution in the least square sense. The theory is first established in the single variable case and then extended to the multivariable framework by means of tensor products. Numerical studies and a real industrial example with a multivariable polynomial regression subject to shape constraints of monotony illustrate the performance of the proposed method.