In this paper, smoothing curve or surface with both interpolation conditions and inequality constraints is considered as a general convex optimization problem in a Hilbert space. We propose a new approximation method based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. An efficient algorithm is derived and numerical examples with bound and monotonicity constraints in one and two dimensions are given. A comparison with existing monotone cubic splines interpolation algorithms in terms of linearized energy criterion is included.