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It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm.
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Boaz Barak, Fernando G.S.L. Brandao, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou, Hypercontractivity, sum-of-squares proofs, and their applications , Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing (New York, NY, USA), STOC ’12, ACM, 2012, pp. 307–326
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Boaz Barak and David Steurer, Sum-of-squares proofs and the quest toward optimal algorithms , In Proceedings of the 2014 International Congress of Mathematicians. International Mathematical Union (2014)
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Gábor Braun, Jonah Brown-Cohen, Arefin Huq, Sebastian Pokutta, Prasad Raghavendra, Aurko Roy, Benjamin Weitz, and Daniel Zink, The matching problem has no small symmetric sdp , Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms (Philadelphia, PA, USA), SODA ’16, Society for Industrial and Applied Mathematics, 2016, pp. 1067–1078
2016
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Ryan O’Donnell, Sos is not obviously automatizable, even approximately , Innovations in Theoretical Computer Science (ITCS) (2017)
2017
Closest in time.