Spectrahedron

Spectrahedron

A spectrahedron is a geometric shape that is defined by a set of linear matrix inequalities and is convex in nature. It is used in semidefinite programming (SDP), which is a type of optimization of a linear function on a spectrahedron. SDP is a broad generalization of linear programming and has many applications in various fields such as control engineering, signal processing, combinatorial optimization, and mechanical structure design.From: Handbook of Graphical Models [2018], SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic [2019]

Gaussian Graphical Models

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Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018

Caroline Uhler

A spectrahedron is a convex set that is defined by linear matrix inequalities. Given a sample covariance matrix S, we define the spectrahedron fiberL(S)={Σ∈S≻0p∣⟨Σ,K⟩=⟨S,K⟩for allK∈L}. $$ \mathrm{fiber}_\mathcal L (S) \quad = \quad \bigl \{ \Sigma \in \mathbb S ^p_{\succ 0} \,\mid \, \langle \Sigma , K \rangle \, = \, \langle S, K \rangle \,\, \text{ for} \text{ all} \,\, K \in \mathcal L \bigr \}. $$

SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic

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Journal Information

Published in Optimization Methods and Software, 2019

Didier Henrion, Simone Naldi, Mohab Safey El Din

Optimization of a linear function on a spectrahedron is called semidefinite programming (SDP), a broad generalization of linear programming with many applications in control engineering, signal processing, combinatorial optimization, mechanical structure design, etc, see [20,22]. The algebra and geometry of spectrahedra is an active area of study in real algebraic geometry, especially in connection with the problem of moments and the decomposition of real multivariate polynomials as sums-of-squares (SOS), see [1,11,17] and references therein. SDP-based methods have recently been developed in the setting of error analysis of roundoff during floating-point computations, see [3,12], or in non-commutative real algebraic geometry, see [4,10].

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Gaussian Graphical Models

SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic