| Project Title: Tensor decompositions for sums of even powers of real polynomials Doctoral Candidate: Matteo Bechere Advisors: S. Kuhlmann (UKON), B. Mourrain (Inria) Mentor: C. Riener (UiT) |
| Objectives: The Archimedean Positivstellensatz for quadratic modules in the multivariate real polynomial ring, coupled with Lasserre’s moment–SOS method, have found important applications in polynomial optimization. The efficiency of the method is based on the fundamental fact that a polynomial is a sum of squares if and only if it admits a positive semidefinite Gram matrix representation, the existence of which can be certified via semidefinite programming. The Archimedean Positivstellensatz holds however in a much greater generality; it was proven by T. Jacobi for arbitrary commutative unital real algebras and their Archimedean sums- of- even- powers modules. The main objectives are to develop an effective method for tensor decompositions of forms as sums of even powers and to exploit the Archimedean Positivstellensatz for sums of even powers modules for polynomial optimisation. This work will be guided by recent progress in Power-Sum Decomposition of Polynomials, where the authors exploit Lasserre’s SOS method to achieve results on higher tensor decompositions. The DC will analysis and then extend the established approaches from sums of squares to sums of even powers of real polynomials, aiming at tractable real tensor decomposition and effective relaxations in polynomial optimisation. |
| Expected results: Achieving an important step towards real algebraic tensor decomposition, and applying it towards a reduction of the size of the corresponding semidefinite programs in polynomial optimisation when working with the smaller cone of sums of even powers instead of the cone of sums of squares. |
| Planned secondment(s): A 10 months secondment (M21-30) to work with B. Mourrain at Inria on real algebraic tensor decomposition followed by a 2 months secondment (M31-32) to work with. C. Riener at UiT on polynomial optimisation aspects. |
| Joint degree: University of Konstanz, Université Côte d’Azur |
