Polynomial OPtimization - POP

Leader : Victor MAGRON
PhD Student Representative : Alexey LAZAREV

POP carries out researches to solve hard nonconvex polynomial optimization problems in all areas of science.
Indeed the last two decades have witnessed the emergence of polynomial optimization as a new field in which powerful positivity certificates from real algebraic geometry based on sums of squares (SOS), representations of measures by their moments, and conic optimization duality, have permitted to develop an original and systematic approach to solve (at global optimality) optimization problems with polynomial (and more generally semi-algebraic) data.

The backbone of this powerful methodology is the "Moment-SOS hierarchy" also known as "Lasserre hierarchy" which has attracted a lot of attention in many areas (e.g., optimization, applied mathematics, quantum computing, engineering, theoretical computer science) with important potential applications.
It has become a basic tool for analyzing hardness of approximation in combinatorial optimization and the best candidate algorithm to prove/disprove Khot's famous Unique Games Conjecture.
Recently it has become a promising new method for solving the important optimal power flow problem (OPF) in the strategic domain of energy networks.

Three books have been recently written on the topics of the Moment-SOS hierarchy, Christoffel-Darboux kernels and sparse polynomial optimization.

Overall, polynomial optimization has a growing inter-disciplinary activity at the intersection of applied mathematics, automatic control and computer science, with several applications of interest, including quantum information theory, deep learning and power systems.
All these activities have benefited from several past/ongoing funded projects at national, European and international levels.