Convex relaxations in circuits, systems, and control

It is the intention of the authors of this paper to provide the reader with a general view of convex relaxations for polynomial optimization problems arising in several research areas, such as systems and control theory, and circuits analysis and design. The paper by no means claims to represent an exhaustive survey on the subject, rather its ambition is to convey to the reader the flavor of a story, where certain abstract pure mathematical results have become the key issue in a wide engineering research area, more than one century after the results were firstly discovered. This is the case of Hilbert’s 17th Problem formulated at the end of the 19th century, asking if a real positive semidefinite multivariate polynomial must necessarily be the sum of squares of rational functions. The strictly related problem asking if a positive semidefinite polynomial can always be represented as the sum of squares of polynomials, was also investigated thoroughly by Hilbert in the decade 1880–1890 and came back in fashion overbearingly in the last 20 years. Actually, a large body of recent work on convex relaxations for nonconvex optimization problems arising in different engineering areas, refers more or less strictly to the above problems. It is the hope of the authors that this review will attract more researchers to these families of optimization problems, both for solving real world problems and advancing the state of the art of the mathematical knowledge base.