  Chordal and factor-width decomposition methods for semidefinite programming
and polynomial optimization have recently enabled the analysis and control of
large-scale linear systems and medium-scale nonlinear systems. Chordal
decomposition exploits the sparsity of semidefinite matrices in a semidefinite
program (SDP), in order to formulate an equivalent SDP with smaller
semidefinite constraints that can be solved more efficiently. Factor-width
decompositions, instead, relax or strengthen SDPs with dense semidefinite
matrices into more tractable problems, trading feasibility or optimality for
lower computational complexity. This article reviews recent advances in
large-scale semidefinite and polynomial optimization enabled by these two types
of decomposition, highlighting connections and differences between them. We
also demonstrate that chordal and factor-width decompositions allow for
significant computational savings on a range of classical problems from control
theory, and on more recent problems from machine learning. Finally, we outline
possible directions for future research that have the potential to facilitate
the efficient optimization-based study of increasingly complex large-scale
dynamical systems.
