We consider the problem of approximating a function from a finite set of its values, provided that the function is subject to constraints. Solving this problem is useful in system identification, state estimation and control. Indeed, in identification and estimation procedures, constrains allow to take into account noise and undermodeling effects. In control, constrained approximation allows online implementation of predictive controllers. The constrained approximation problem is approached by means of the nonlinear set membership (NSM) method. The main feature of this method is that no assumptions on the parametric form of the function to approximate are used. Only regularity assumptions on the function are taken. In this way, the complexity/accuracy problems posed by the choice of the parametrization are avoided. In this paper, a method providing optimal constrained approximation is derived. The method is applied to fast implementation of model predictive control.