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Equivariant semidefinite lifts of regular polygons
(Institute for Operations Research and the Management SciencesMathematics of Operations Research, 2017-05-01)
Given a polytope P in $\mathbb{R}^n$, we say that P has a positive
semidefinite lift (psd lift) of size d if one can express P as the linear
projection of an affine slice of the positive semidefinite cone
$\mathbf{S}^d_+$. ...
Lieb's concavity theorem, matrix geometric means, and semidefinite optimization
(Linear Algebra and Its Applications, 2017-01-15)
A famous result of Lieb establishes that the map (A,B)↦tr[K^* A^{1−t}KB^{t}] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t∈[0,1]. In this paper we show that Lieb's ...
A lower bound on the positive semidefinite rank of convex bodies
The positive semidefinite rank of a convex body $C$ is the size of its
smallest positive semidefinite formulation. We show that the positive
semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$
is ...