Spectrahedron
In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of n × n positive semidefinite matrices forms a convex cone in Rn × n, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace.
Spectrahedra are the feasible regions of semidefinite programs.[1] The images of spectrahedra under linear or affine transformations are called projected spectrahedra or spectrahedral shadows. Every spectrahedral shadow is a convex set that is also semialgebraic, but the converse (conjectured to be true until 2017) is false.[2]
An example of a spectrahedron is the spectraplex, defined as
- [math]\displaystyle{ \mathrm{Spect}_n = \{ X \in \mathbf{S}^n_+ \mid \operatorname{Tr}(X) = 1\} }[/math],
where [math]\displaystyle{ \mathbf{S}^n_+ }[/math]is the set of n × n positive semidefinite matrices and [math]\displaystyle{ \operatorname{Tr}(X) }[/math] is the trace of the matrix [math]\displaystyle{ X }[/math].[3] The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.
See also
- N-ellipse - a special case of spectrahedra.
References
- ↑ Ramana, Motakuri; Goldman, A. J. (1995), "Some geometric results in semidefinite programming", Journal of Global Optimization 7 (1): 33–50, doi:10.1007/BF01100204.
- ↑ Scheiderer, C. (2018-01-01). "Spectrahedral Shadows". SIAM Journal on Applied Algebra and Geometry 2: 26–44. doi:10.1137/17m1118981.
- ↑ Gärtner, Bernd; Matousek, Jiri (2012). Approximation Algorithms and Semidefinite Programming. Springer Science and Business Media. pp. 76. ISBN 978-3642220159. https://archive.org/details/approximationalg00grtn_585.
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