The totally nonnegative completion problem
An n-by-n real matrix is said to be totally positive (nonnegative) if every minor (principal and non-principal) is positive (nonnegative). The totally nonnegative completion problem asks which partially totally nonnegative matrices have a completion to a totally nonnegative matrix. Here we settle the first natural question: for which (labeled) graphs G does every partial totally nonnegative matrix, the graph of whose specified entries is G, have a totally nonnegative completion? Just as in the positive definite case this must play a key role in any further development of the theory.
Topics in Semidefinite and Interior-Point Methods
Johnson, C. R., Kroschel, B. K., & Lundquist, M. (1998). The totally nonnegative completion problem. In P. M. Pardalos & H. Wolkowicz (Eds.), Fields Institute Communications: Topics in Semidefinite and Interior-Point Methods. AMS. https://doi.org/10.1090/fic/018/07