Logiciels, Prototypes et Démonstrateurs - POP
The team members have been implementing several software packages with open-source tools, released under free licenses. Advantages of this policy include the free access to source-code information, collaboration across academic partners without compromising the design of quality software and content for peer-reviewing. We present below a non-exhaustive list of these software libraries.
GloptiPoly3: Moments, optimization and semidefinite programming. Matlab parser for generalized problems of moments. Allows to build and solve convex linear matrix inequality (LMI) relaxations of the (generally non-convex) global optimization problem of minimizing a multi-variable polynomial function subject to polynomial inequality, equality or integer constraints. Can be freely downloaded and used. Developed by/with the help of: Didier Henrion, Jean-Bernard Lasserre and Johan Löfberg (Linköping University).
TSSOS: This is an open source Julia library developed for large-scale polynomial optimization, based on the sparsity adapted moment-SOS hierarchies. Related modules can perform complex polynomial optimization, eigenvalue/trace optimization of noncommutative polynomials, compute joint spectral radii for stability analysis, approximate attractors and invariants of sparse dynamical systems. Developed by/with the help of: Victor Magron, Jie Wang (Chinese Academy of Sciences, Beijing).
Validated computing software
We propose efficient numerical routines together with sure and reasonably tight error bounds. Applications include computer aided proofs, certification of numerical results in safety-critical control systems (e.g. cyber-physical systems).
RealCertify: Maple package to tackle the problem of deciding the non-negativity of rational polynomials over semi-algebraic domains defined by polynomial constraints with rational coefficients.
This is done by computing sum of squares certificates of non-negativity for inputs where such certificates hold over the rational numbers.
It can be applied to numerous problems coming from engineering sciences, program verification and cyber-physical systems.
It is based on hybrid symbolic-numeric algorithms relying on semi-definite programming. Developed by/with the help of: Victor Magron, Mohab Safey El Din (Sorbonne University).
SPECTRA: a Maple library which aims at finding at least one point in a spectrahedron, using exact arithmetic. SPECTRA should be used for small-dimension problems. It should not be considered as a competitor to numerical algorithms such as interior-point methods for semidefinite programming (SDP).
Contrary to numerical algorithms which are based on approximate computations and floating point arithmetic, SPECTRA is exclusively based on computations with exact arithmetic, and hence it should be primarily used either in potentially degenerate situations, for example when it is expected that the spectrahedron has empty interior, or when a rigorous certificate of infeasibility or feasibility is required.
Developed by/with the help of: Didier Henrion, Simone Naldi (XLIM Limoges), Mohab Safey El Din (Sorbonne University).