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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.

14P10 Semialgebraic sets and related spaces
90C22 Semidefinite programming
14N05 Projective techniques in algebraic geometry
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