Abstract
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.
Highlights
The design of innovative technology capable to address the challenges of the 21st century relies on the ability to analyze, predict, and control large-scale complex systems, which are typically nonlinear and may interact over networks (Astrom & Kumar, 2014; Murray et al, 2003)
Convex optimization is one of the key tools for achieving these goals, because many questions related to the stability and operational safety of dynamical systems, the synthesis of optimal control policies, and the certification of robust performance can be posed as convex optimization problems
These take the form of semidefinite programs (SDPs)—linear optimization problems with positive semidefinite matrix variables
Summary
Convex optimization is one of the key tools for achieving these goals, because many questions related to the stability and operational safety of dynamical systems, the synthesis of optimal control policies, and the certification of robust performance can be posed as (or relaxed into) convex optimization problems Very often, these take the form of semidefinite programs (SDPs)—linear optimization problems with positive semidefinite matrix variables. As emphasized in the previous sections, solving large-scale semidefinite programs is at the centre of many problems in control engineering and beyond, and the development of fast and reliable solvers has attracted significant attention recently, mainly focusing on sparsity exploiting and low-rank solution exploiting methods (De Klerk, 2010; Majumdar et al, 2020).
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