bordered Hessian

The Lagrange method finds candidate points where the objective and constraint gradients are parallel, but it stays silent on whether each candidate is a constrained maximum, minimum, or neither. The bordered Hessian is the second-order test that settles the question — the constrained analogue of checking the definiteness of the ordinary Hessian.

It is built by taking the Hessian of the Lagrangian (the second derivatives with respect to the variables) and surrounding it with a border made from the gradient of the constraint. For one constraint g in n variables, the matrix is [0, grad g; (grad g) transpose, H], an (n+1) by (n+1) symmetric matrix. The trick is that ordinary definiteness is the wrong question — you only care about curvature along directions that respect the constraint, the tangent space. The border encodes exactly that restriction, and the test reads the signs of a particular sequence of its leading principal minors. The right alternating pattern of signs signals a constrained minimum; the opposite pattern, a constrained maximum.

In economics this test confirms that a cost-minimizing or utility-maximizing solution really is optimal, not just stationary. It is the rigorous backstop behind second-order conditions in constrained optimization and the basis for sensitivity analysis. The honest caveat: the sign-of-minors rule is finicky, the exact pattern depends on the number of constraints, and in practice one often instead checks the eigenvalues of the Hessian restricted to the tangent space, which is the same idea stated more transparently.