Global optimization for low-dimensional switching linear regression and bounded-error estimation

TL;DR

This paper introduces globally optimal algorithms for hybrid system identification problems, specifically switching linear regression and bounded-error estimation, ensuring solutions with certificates of global optimality.

Contribution

It presents a branch-and-bound approach with efficient lower bounds, enabling scalable, globally optimal solutions without integer variables.

Findings

Higher accuracy than convex relaxations

Reasonable computational burden for hybrid system identification

Promising results in robust estimation with outliers

Abstract

The paper provides global optimization algorithms for two particularly difficult nonconvex problems raised by hybrid system identification: switching linear regression and bounded-error estimation. While most works focus on local optimization heuristics without global optimality guarantees or with guarantees valid only under restrictive conditions, the proposed approach always yields a solution with a certificate of global optimality. This approach relies on a branch-and-bound strategy for which we devise lower bounds that can be efficiently computed. In order to obtain scalable algorithms with respect to the number of data, we directly optimize the model parameters in a continuous optimization setting without involving integer variables. Numerical experiments show that the proposed algorithms offer a higher accuracy than convex relaxations with a reasonable computational burden for hybrid system identification. In addition, we discuss how bounded-error estimation is related to robust estimation in the presence of outliers and exact recovery under sparse noise, for which we also obtain promising numerical results.