Spectrahedron

A spectrahedron

In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of $n \times n$ positive semidefinite matrices forms a convex cone in $R n \times n$ , and a spectrahedron is a shape that can be formed by intersecting this cone with a linear affine subspace.

Spectrahedra are the solution spaces of semidefinite programs.^[1] Every spectrahedron is a convex set and also semialgebraic, but the reverse (conjectured to be true until 2017) is false.^[2]

An example of a spectrahedron is the spectraplex, defined as

$\mathrm {Spect} _{n}=\{X\in \mathbf {S} _{+}^{n}\mid \mathrm {Tr} (X)=1\}$

where $\mathbf {S} _{+}^{n}$ is the set of $n \times n$ positive semidefinite matrices and $\mathrm {Tr} (X)$ is the trace of the matrix $X$ .^[3] The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.

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: 26–44. doi:10.1137/17m1118981.
↑ Gärtner, Bernd; Matousek, Jiri (2012). Approximation Algorithms and Semidefinite Programming. Springer Science and Business Media. p. 76. ISBN 3642220150.

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[1] Ramana, Motakuri; Goldman, A. J. (1995), "Some geometric results in semidefinite programming", Journal of Global Optimization, 7 (1): 33–50, doi:10.1007/BF01100204 .

[2] Scheiderer, C. (2018-01-01). "Spectrahedral Shadows". SIAM Journal on Applied Algebra and Geometry: 26–44. doi:10.1137/17m1118981.

[3] Gärtner, Bernd; Matousek, Jiri (2012). Approximation Algorithms and Semidefinite Programming. Springer Science and Business Media. p. 76. ISBN 3642220150.