For my secondment, I moved to Toulouse to join the polynomial optimization group at CNRS-LAAS. As someone working in polynomial optimization, Toulouse is a particularly exciting place to do research: the subject has a long history there, with Jean Bernard Lasserre’s work on moment and sum-of-squares methods having played a foundational role in the development of the field. My stay gave me the opportunity to concentrate on some of the more theoretical aspects of polynomial optimization and, at the same time, to exchange ideas with researchers working on closely related topics.
A central topic of my secondment was the question of how to extract global minimizers from semidefinite relaxations of polynomial optimization problems. In the classical moment-SOS hierarchy, so-called flatness conditions provide a powerful certificate: under suitable rank conditions on the moment matrix, one can certify that the relaxation is already exact and recover the global minimizers. Together with Victor Magron, we investigated how this theory changes when one exploits structure to reduce the size of the semidefinite problems. We developed a block flatness condition that plays an analogous role when the moment matrix decomposes into smaller blocks. This result can, in particular, be applied to term sparsity based relaxations, providing a practical certificate of optimality together with a procedure for recovering minimizers from the reduced semidefinite program.
We also investigated the closely related situation when symmetries of the polynomial optimization problem are exploited. In this setting, we established a connection between flatness of the invariant moment block and the so-called orbit space reduction technique, providing another route from a reduced semidefinite relaxation back to the minimizers of the original problem. Beyond the theoretical results, an important next step is implementation: together with our collaborators, we plan to incorporate these extraction procedures into TSSOS, a Julia package for solving large scale structured polynomial optimization problems.
One particularly enjoyable aspect of the secondment was the working environment at LAAS. I shared an office with two other TENORS PhD students, Llorenç and Younes, both physicists working on problems related to quantum theory. Coming from a more optimization oriented background myself, I found it particularly interesting to see how they approached mathematical problems and which questions they considered natural. Many discussions started casually in the office and our many informal discussions were a valuable part of the stay.
Looking back, the secondment in Toulouse was valuable both mathematically and personally. It gave me the time and environment to focus on theoretical questions, which developed into concrete results and ongoing projects. At the same time, working alongside researchers with rather different backgrounds broadened my perspective on how optimization techniques can interact with other areas, particularly quantum information. I am looking forward to continuing the collaborations and projects that grew out of my time at LAAS.