We show that the second–order condition for strict local extrema in both constrained and unconstrained optimization problems can be expressed solely in terms of principal minors of the (Lagrengean) [sic] Hessian. This approach unifies the determinantal tests in the sense that the second-order condition can be always given solely in terms of Hessian matrix.
Im, E.I. (2005), Hessian sufficiency for bordered Hessian,
Research Letters in the Information and Mathematical Sciences, 8, 189-196