Monotonically Constrained Polynomial Regression: An Application of Sum of Squares Techniques and Semidefinite Programming

Abstract

This paper proposes a procedure for incorporating strict monotonicity constraints. We develop an algorithm for constraining monotonicity of polynomial multivariate functions on compact subsets of $\mathbb{R}^n$. MCPR (monotonically constrained polynomial regression) is modeled as a constrained Sum of Squares optimization problem which can be solved as a Semidefinite Program (SDP). We show that MCPR can approximate arbitrarily well any function that satisfies given monotonicity constrains on a compact set. We find that in some scenarios MCPR performs better than "state of the art" algorithms, such as Neural Networks and Regression Trees. However, MCPR is computationally expensive and more research is necessary in order to improve its scalability and make it applicable to high dimensional frameworks.