Sparse Identification of Polynomial and Posynomial Models

Posynomial models are widely used in various engineering design endeavors, such as circuits, aerospace and structural design, mainly due to the fact that design problems cast in terms of posynomial objectives and constraints can be solved efficiently by means of a convex optimization technique known as geometric programming (GP). However, while quite a vast literature exists on GP-based design, very few contributions can yet be found on the problem of identifying posynomial models from experimental data. Posynomials are nonnegative combinations of monomials with possibly fractional and both positive and negative exponents. Thus, posynomial identification amounts to determining not only the coefficients of the combination, but also the exponents in the monomials, which renders the identification problem numerically hard. In this preliminary draft, we outline the idea of an approach to the identification of both multivariate polynomial and posynomial models, based on the expansion on a given large-scale basis of monomials. The model is then identified by seeking coefficients of the combination that minimize a mixed objective, composed by a term representing the fitting error and a term inducing sparsity in the representation, which result in a problem formulation of the “square-root LASSO” type, with nonnegativity constraints on the variables. We propose to solve the problem via a sequential coordinate-descent scheme, which is suitable for large-scale implementations.