From Simon and Blume (p. 391):
Using a bordered hessian H with m constraints, to verify positive definiteness, check that det(H2m+1) has the same sign as (-1)m and that all larger leading principal minors have this sign too.
To verify negative definiteness, check that det(H2m+1) has the same sign as (-1)m+1 and that the leading principal minors of larger order alternate in sign.
I wanted to know if there are similar rules in checking for positive/negative semi-definiteness, since the objective function need only be quasiconcave/quasiconvex for my purposes. Is there a resource with as clean notation as this?
Yes, there are. You have to look at all the principal minors, not just the leading principal minors. See the old book by W.L. Ferrar, Algebra published by Oxford U.P.