I am interested in computational approaches to problems involving systems of polynomial equations and inequalities, e.g., to the general problem of polynomial optimization. This is why I implement algorithms in computer algebra systems such as Maple or Macaulay2.
Semidefinite Programming solved Exactly with Computational Tools of Real Algebra
Image 1: a spectrahedron Image 2: sample points on a quartic curve
SPECTRA is a Maple library for solving linear matrix inequalities (LMI) in exact arithmetic. An example of solution set of a LMI is the red convex region of Image 1, or the interior of the inner oval of Image 2.
Please visit the Spectra permanent webpage for further information.
D. Henrion, S. Naldi, M. Safey El Din. Spectra: a Maple library for solving linear matrix inequalities in exact arithmetic. Software documentation, November 2016 [LINK].
An example of quintic spectrahedron of type (12,0) computed by Spectra : quintic.
Optimization with hyperbolic polynomials
Image 3: Hyperbolic quartic Image 4: Hyperbolic quintic
This is a beta-version of a library for exact computing with hyperbolic polynomials.
Binary file : HYPER
This library implements algorithms presented in:
- S. Naldi, D. Plaumann. Symbolic computation in hyperbolic programming. Accepted for oral presentation at MEGA 2017, Nice, Juin 2017. Submitted for publication (2016). [arxiv]