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Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate approximate solutions in floating point arithmetic. In the first part of the course we describe semidefinite programming… (More)

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications… (More)

Let <i>H</i><sub>0</sub>, …, <i>H <sub>n</sub></i> be <i>m</i> x <i>m</i> matrices with entries in <i>Q</i> and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix <i>H</i>(x) = <i>H</i><sub>0</sub>+<i>x</i><sub>1</sub><i>H</i>_1+…+<i>x<sub>n</sub>H<sub>n</sub></i> and the problem of computing sample points in… (More)

This document briefly describes our freely distributed Maple library spectra, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.

Let A 0 , A 1 ,. .. , A n be given square matrices of size m with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety {x ∈ R n : det(A 0 + x 1 A 1 + · · · + x n A n) = 0}. Such a problem finds applications in many areas such as control theory, computational geometry,… (More)

Let A(x) = A 0 + x 1 A 1 + · · · + x n A n be a linear matrix, or pencil, generated by given symmetric matrices A 0 , A 1 ,. .. , A n of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semi-algebraic set called spectrahedron, described by a linear matrix inequality (LMI). We design an exact… (More)

- Simone Naldi
- 2015

The problem of finding low rank m × m matrices in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parcimony. We design a symbolic computation algorithm to solve this problem efficiently, exactly and rigorously: the input are the rational coefficients of the matrices… (More)

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