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Generic spectrahedral shadows. (English) Zbl 1346.14132
Spectrahedral sets (or sets admitting an LMI representation) are linear sections of the cone of positive semidefinite matrices. A spectrahedral shadow is a (closed convex semialgebraic) set \(S\subset \mathbb{R}^{d}\) that can be represented as a projection to the first \(d\) coordinates of some spectrahedral subset of \(\mathbb{R}^{d+p}\), that is, \[ S=\left\{ x\in \mathbb{R}^{d}\mid \exists y\in \mathbb{R}^{p}: \,\sum_{i=1}^{d}x_{i}A_{i}+\sum_{j=1}^{p}y_{j}B_{j}+C\succeq 0\right\}, \] where \(A_i,B_j,C\) are real symmetric \(n\times n\) matrices. The question of whether or not every closed convex semialgebraic set is a spectrahedral shadow remains widely open. In this work, the authors study the structure of the boundary of spectrahedral shadows of type \((n,d,p)\) (referring to the dimension of matrices, and respectively the Euclidean spaces involved in the definition of \(S\)), under the assumption that both the linear section (defining the spectrahedral shadow in \(\mathbb{R}^{d+p}\)) and the projection to \(\mathbb{R}^{d}\) are generic. In this case, they characterize polynomials vanishing in the boundary of \(S\) (Theorem 1.1). They also discuss several examples of spectrahedral shadows in dimensions 2 (mostly) and 3. The authors fairly mention in the last section that the genericity assumption made on the spectahedral shadows compromises applications to optimization, at least in the current stage.

MSC:
14P10 Semialgebraic sets and related spaces
90C22 Semidefinite programming
14N05 Projective techniques in algebraic geometry
Software:
Macaulay2
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References:
[1] D. Amelunxen and P. Bürgisser, Intrinsic volumes of symmetric cones, Math. Program. Ser. A, 149 (2015), pp. 105–130. · Zbl 1311.90092
[2] G. Blekherman, P. A. Parrilo, and R. R. Thomas, eds., Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Ser. Optim. 13, SIAM, Philadelphia, 2013.
[3] H.-C. Graf von Bothmer and K. Ranestad, A general formula for the algebraic degree in semidefinite programming, Bull. Lond. Math. Soc., 41 (2009), pp. 193–197. · Zbl 1185.14047
[4] D. Grayson and M. Stillman, Macaulay\textup2, a Software System for Research in Algebraic Geometry and Commutative Algebra, Department of Mathematics, University of Illinois, Urbana, IL, 2014. Available online at http://www.math.uiuc.edu/Macaulay2/.
[5] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York, 1977.
[6] W. Helton and J. Nie, Semidefinite representation of convex sets, Math. Program. Ser. A, 122 (2010), pp. 21–64. · Zbl 1192.90143
[7] J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming, Math. Program. Ser. A, 122 (2010), pp. 379–405. · Zbl 1184.90119
[8] J. C. Ottem, K. Ranestad, B. Sturmfels, and C. Vinzant, Quartic spectrahedra, Math. Program. Ser. B, 151 (2015), pp. 585–612.
[9] M. Ramana and A. J. Goldman, Some geometric results in semidefinite programming, J. Global Optim., 7 (1995), pp. 33–50. · Zbl 0839.90093
[10] P. Rostalski and B. Sturmfels, Dualities, in Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Ser. Optim. 13, G. Blekherman, P. A. Parrilo, and R. R. Thomas, eds., SIAM, Philadelphia, 2013, pp. 203–249. · Zbl 1360.90195
[11] C. Scheiderer, Semidefinite Representations for Convex Hulls of Real Algebraic Curves, preprint, arXiv:1208.3865v3 [math.AG], 2012.
[12] R. Sinn, Algebraic boundaries of convex semi-algebraic sets, Res. Math. Sci., 2 (2015), 3. Available online at http://www.resmathsci.com/content/2/1/3. · Zbl 1381.52007
[13] B. Sturmfels and C. Uhler, Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry, Ann. Inst. Statist. Math., 62 (2010), pp. 603–638. · Zbl 1440.62255
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