Toward Tractable Global Solutions to Maximum-Likelihood Estimation Problems via Sparse Sum-of-Squares Relaxations

In system identification, the maximum-likelihood method is typically used for parameter estimation owing to a number of optimal statistical properties. However, in many cases, the likelihood function is nonconvex. The solutions are usually obtained by local numerical optimization algorithms that require good initialization and cannot guarantee global optimality. This paper proposes a computationally tractable method that computes the maximum-likelihood parameter estimates with posterior certification of global optimality via the concept of sum-of-squares polynomials and sparse semidefinite relaxations. It is shown that the method can be applied to certain classes of discrete-time linear models. This is achieved by taking advantage of the rational structure of these models and the sparsity in the maximum-likelihood parameter estimation problem. The method is illustrated on a simulation model of a resonant mechanical system where standard methods struggle.