ANR PROJECT HYPERSPACE
Title of the project:
HYPERSPACE
Principal investigator:
Simone Naldi (Maître de conférences, Université de Limoges, France)
Period:
01/01/2022 — 31/12/2025
Keywords:
Computer algebra
Complexity of semidefinite and hyperbolic programming
Determinantal and spectrahedral representations
Certificates of hyperbolicity
Hyperbolic relaxations


Abstract:
In the project HYPERSPACE we develop effective methods, algorithms and software dedicated to hyperbolic polynomials. These form a class of real multivariate polynomials of central interest in applied mathematics and computer science (because of their importance in polynomial optimization and control theory, just to name two applications). The major open question related to this theory is the socalled Generalized Lax Conjecture, concerned with the representability of hyperbolic polynomials by means of determinants of symmetric matrices. More generally, a challenge in terms of complexity is the problem of certifying whether a given multivariate polynomial is hyperbolic, in other words, the hyperbolicity test.
Efficient algorithms and good complexity bounds as well as software implementations for hyperbolic polynomials are missing but considered crucial for the future development of this theory. The goal of our project HYPERSPACE is twofold: on the one hand we want to develop new effective methods and algorithms for hyperbolic polynomials, on the other hand we aim at implementing reliable software as a support to current and future research.
Previous related work:
M. Kummer, S. Naldi and D. Plaumann. Spectrahedral representations of plane hyperbolic curves, Pac. J. Math. 303(1):243263 (2019)
S. Naldi and D. Plaumann. Symbolic computation in hyperbolic programming, J. Algebra Appl. 17:10 (2018)

Activities funded by the ANR HYPERSPACE:
Invitation of V. Neiger (Lip6, Sorbonne Univ.) at XLIM (Limoges) on 1721/01/2022
Organisation of the workshop Structured Matrix Days at XLIM (Limoges) on 1920/05/2022 (cofunded by Instit. XLIM and GdR IM)
Sponsor:
