 Reduced Hessian

For equality-constrained optimization, restrict the Hessian matrix $H$ of the optimization Lagrangian to feasible tangent directions. If the columns of $Z$ form a basis of the nullspace of the constraint Jacobian matrix $A$, the resulting quadratic form has matrix $Z^{T}HZ$. Positive definiteness gives positive quadratic curvature in every nonzero tangent direction. Nonsingularity of this reduced matrix, together with full row rank of $A$, characterizes nonsingularity of the corresponding equality-constrained saddle-point matrix.