Many applications, including computer vision, computer arithmetic, deep learning, entanglement in quantum information, graph theory and energy networks, can be successfully tackled within the framework of polynomial optimization, an emerging field with growing research efforts in the last two decades. One key advantage of these techniques is their ability to model a wide range of problems using optimization formulations. Polynomial optimization heavily relies on the moment-sums of squares (moment-SOS) approach proposed by Lasserre, which provides certificates for positive polynomials. On the practical side, however, there is "no free lunch" and such optimization methods usually encompass severe scalability issues. Fortunately, for many applications, including the ones formerly mentioned, we can look at the problem in the eyes and exploit the inherent data structure arising from the cost and constraints describing the problem. This book presents several research efforts to resolve this scientific challenge with important computational implications. It provides the development of alternative optimization schemes that scale well in terms of computational complexity, at least in some identified class of problems. It also features a unified modeling framework to handle a wide range of applications involving both commutative and noncommutative variables, and to solve concretely large-scale instances. Readers will find a practical section dedicated to the use of available open-source software libraries. This interdisciplinary monograph is essential reading for students, researchers and professionals interested in solving optimization problems with polynomial input data.

�ber die Autorin bzw. den Autor:

Victor Magron is a full-time researcher at CNRS-LAAS, France, working in the MAC team. He completed his PhD in computer science in 2013 at Ecole Polytechnique, INRIA-Saclay, and defended his habilitation thesis in 2021. In 2014, he was a postdoc in the MAC team. In 2014–2015, he was a research associate in the Circuits and Systems group at Imperial College. From 2015 to 2018, he was a CNRS junior researcher affiliated to the Tempo team at Verimag in Grenoble. In 2018, he visited the joint INRIA-CNRS-Sorbonne Universit� PolSys team at LIP6 in Paris Jussieu. His research is devoted to applications of certified polynomial optimization to deep learning, quantum and power systems. He has published 50 peer-reviewed articles.

Jie Wang is an associate research fellow at Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), China. He completed his PhD in mathematics in 2017 at Academy of Mathematics and Systems Science, CAS. In 2017–2019, he was a postdoc at Peking University. In 2019–2021, he was a postdoctoral researcher at CNRS-LAAS. He works in the areas of polynomial optimization, semidefinite programming, real algebraic geometry, symbolic computation and their applications in control, quantum information, computer vision and so on. He has published 20 peer-reviewed articles.

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