by Sebastien Debus, UiT The Arctic University of Norway
As part of my POEMA project, I have the opportunity to complete two secondments in France. Both secondments last two months. I spend January and April 2022 at Inria Sophia Antipolis, where I am hosted by Evelyne Hubert. In February and March, I am at the company Artelys in Paris and am hosted by Michaël Gabay.
At Inria I had several exchanges with Evelyne Hubert and Tobias Metzlaff in January. I enjoyed hearing about their research. Their recent preprint “Polynomial description for the T-Orbit Spaces of Multiplicative Actions” deals with invariant polynomials under the multiplicative action of Weyl groups. This is a fascinating topic and intersects with the research I have previously done. For reduced Weyl groups of the types A, B, C, D they provide a description of the orbit space as in the classical work of Procesi and Schwarz “Inequalities defining orbit spaces”.
Inria – Sophia Antipolis
In my research I studied applications of the linear action of real reflection groups in real algebraic geometry and optimization. A particular interest was describing the equivariants. Although the groups A, B, C, D are as groups isomorphic to essential reflection groups, the representation theory of Weyl groups differs, as one acts on the Laurent polynomial ring. I learned from Evelyne and Tobias about the topic. Furthermore, Tobias and I started research on the following: Given Weyl groups A, B, C, D what is the isotypic decomposition of the coinvariant algebra and what structure does it have? We tried to adapt Higher Specht polynomial to the Weyl groups. Our work has computational and algebraical components. In January, we made progress with the groups of type B and look forward to proceeding with our joint work in April. Hopefully, we will be able to complete our work with a research article at some point.
Artelys is an independent company specialised in optimization, decision support and modeling
At Artelys, I work on concrete problems in optimization. Ahmadi and Hall started to study how well linear programming can be used to solve SDPs. The authors propose to consider instead of the SDP cone the cone of all diagonal dominant matrices (DD) which can be formulated via linear constraints. They also suggest considering a SOCP via the notion of scaled diagonal dominant matrices (SDD). Although the cone of DD matrices has finitely many extremal rays, it seems that it can be used well to solve some SDP problems. Thus, the cone of DD matrices is a polyhedra and well studied.
Felix Kirschner started working on this project during his secondment at Artelys. The idea is to obtain some practical information about the usefulness of the approach via comparing the proposed LP and SOCP problems to the SDP of some well known (combinatorial) problems (Lovász number, set partition problem and sums of squares testing). Felix implemented the dual formulation for a matrix being DD and I implemented the primal formulation. It seems that both formulations function similarly well. The DD approach works well on the Lovász number, but is not very successful for the set partition problem. A reason could be that if the multiset is not partitionable (i.e. some polynomial f_A is strictly positive), the maximum value e for which f_A - e is sum of squares is very small. I implement also the SDD formulation and do some comparison tests.
