We propose a new method for estimating a regression function from noisy data when the underlying function is known to satisfy a certain smoothness condition. The proposed method fits a function to the data set so that the roughness of the fitted function is minimized while ensuring that the sum of the absolute deviations of the fitted function from the data points does not exceed a certain limit. It is shown that the fitted function exists and can be computed by solving a quadratic program. Numerical results demonstrate that the proposed method generates more efficient estimates than its alternative in terms of the mean square error and the amount of time required to compute the fit.