# Optimal Size of Linear Matrix Inequalities in Semidefinite Approaches to Polynomial Optimization

@article{Averkov2018OptimalSO, title={Optimal Size of Linear Matrix Inequalities in Semidefinite Approaches to Polynomial Optimization}, author={Gennadiy Averkov}, journal={SIAM J. Appl. Algebra Geom.}, year={2018}, volume={3}, pages={128-151} }

The abbreviations LMI and SOS stand for `linear matrix inequality' and `sum of squares', respectively. The cone $\Sigma_{n,2d}$ of SOS polynomials in $n$ variables of degree at most $2d$ is known to have a semidefinite extended formulation with one LMI of size $\binom{n+d}{n}$. In other words, $\Sigma_{n,2d}$ is a linear image of a set described by one LMI of size $\binom{n+d}{n}$. We show that $\Sigma_{n,2d}$ has no semidefinite extended formulation with finitely many LMIs of size less than…

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