Consistency of Multi-dimensional Convex Regression
E. Lim and P. W. Glynn
Operations Research, Vol. 60, 196-208 (2012)
Convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is (possibly) multi-dimensional. Such regression problems arise in operations research, economics, and other disciplines in which imposing a convexity constraint on the regression function is natural. This paper studies a least squares estimator that is computable as the solution of a quadratic program and establishes that it converges almost surely to the “true” function as 
under modest technical assumptions. In addition to this multi-dimensional consistency result, we identify the behavior of the estimator when the model is mis-specified (so that the “true” function is non-convex) and extend the consistency result to settings in which the function must be both convex and non{decreasing (as is needed for consumer preference utility functions).