Convex version of multivariate adaptive regression splines for optimization

Multivariate Adpative Regression Splines (MARS) provide a flexible statistical modeling method that employs forward and backward search algorithms to identify the combination of basis functions that best fits the data. In optimization, MARS has been used successfully to estimate the value function in stochastic dynamic programming, and MARS could be potentially useful in many real world optimization problems where objective (or other) functions need to be estimated from data, such as in simulation optimization. Many optimization methods depend on convexity, but a nonconvex MARS approximation is inherently possible because interaction terms are products of univariate terms. In this paper, we propose a convex version of MARS. In order to ensure MARS convexity, two major modifications are made: (1) coefficients are constrained such that pairs of basis functions are guaranteed to jointly form convex functions; (2) The form of interaction terms is appropriately changed. Finally, MARS convexity can be achieved by the fact that the sum of convex functions is convex.