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JMI2013A-1 Remarks on positivity of $\alpha$-determinants via SDP relaxation (pp.1-10)

Author(s): Takayuki Osogami, Tomoyuki Shirai and Hayato Waki

J. Math-for-Ind. 5A (2013) 1-10.

After we survey our positivity problem for $\alpha$-determinants of matrices, we reformulate it as polynomial optimization problems. By using SDP relaxation, we perform numerical computation for these optimization problems. We also give SOS representations for the $\alpha$-determinants to obtain the optimal value when the matrix size $n=3$, and give conjectures for $n=4,5$.

Keyword(s).  $\alpha$-determinant, determinant, permanent, point process, positivity, SDP relaxation, SOS representation