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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
Macaulay2
Full Text:
##### 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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